The atomic nucleus is formed of neutrons and protons, which are in turn formed of "up" and "down" quarks. These quarks are confined within the neutrons and protons, and held together by gluons. The attractive force of these gluons is millions of times greater than that of electromagnetism; in fact, it's so strong that it's the strongest force we know of, and it is called the "strong nuclear force" for that reason. A quantum theory of how this force operates has been devised, which is called "Quantum Chromo-Dynamics," or "QCD" for short.

The "up" quark is the lightest quark; the "down" is heavier. Therefore, there is a way that the down quark can decay into an up quark. The force that drives this decay is called the "weak nuclear force." There is also a quantum theory for how the weak force operates; however, unlike the strong force, the weak force is apparently associated in a deep manner with electromagnetism. This quantum theory is therefore called "electro-weak theory."

Protons are not made just of up quarks; they are made of two ups and a down, whereas neutrons are made of two downs and an up. Apparently, up quarks are the most stable, but they can't exist without at least one down. Quarks come two ways: in pairs, and in trios. The pairs are unstable, and consist of one quark and one antiquark of another flavor ("up" and "down" are called "flavors," for historical reasons). This is because if they were the same flavor, they would annihilate and become energy, just as antiprotons can with protons, or electrons can with positrons. The trios, on the other hand, are either all quarks or all antiquarks, and can be stable; but the only truly stable trio is the proton, two ups and a down. The simplest pair is an up and an antidown; this is called a "pion," and it is the simplest and lightest of the particles made from quarks (electrons are not made from quarks, they are stable all by themselves). The pion is unstable, decaying in a very short time (millionths of a second) by a weak force interaction into a muon (which is to an electron as another type of quark, the "strange" quark, is to the up quark) and a muon antineutrino (which is related to the muon as the down quark is to the up quark).

The neutron, by itself, is also unstable; it decays by a weak force interaction into a proton, an electron, and an electron antineutrino. The neutron, however, is much longer-lived than the pion; about 15 minutes. In sub-atomic terms, where significant events take place in millionths or billionths of a second, or even shorter times, this might as well be forever. So it is fair to say that the neutron is "almost stable."

It is worth stopping a moment to mention the elementary particles and their relations to one another. Here is a list:

Leptons
Electron: mass about 1/2MeV, electrically charged negative, spin 1/2 (fermion), interacts with gravity and electromagnetism but not with the strong or weak forces; stable.
Muon: mass about 106MeV, electrically charged negative, spin 1/2 (fermion), interacts with gravity and electromagnetism and the weak force, but not the strong force; unstable, decays in about 2.2 microseconds into an electron and electron-antineutrino.
Tau: mass about 1777MeV, electrically charged negative, spin 1/2 (fermion), interacts with gravity and electromagnetism and the weak force, but not the strong force; unstable, decays in about 3x10^-13 seconds into a muon and muon-antineutrino.
Neutrinos: There are three neutrinos, one each for the electron, muon, and tau. They are currently believed to have a very small but non-zero mass, and current ideas indicate that they may fluctuate from one flavor to another as they propagate, although the evidence for these two beliefs remains somewhat equivocal. They have spin 1/2 (fermion), and interact only with the weak force (and gravity, if in fact it turns out that they do have mass).
Quarks
Up: Mass between 1.5 and 4.0MeV, depending on how it is measured; electrically charged +2/3, spin 1/2 (fermion), interacts via the strong, electromagnetic, and gravitational forces, but not weak; stable.
Down: Mass between 4 and 8MeV, electrically charged -1/3, spin 1/2 (fermion), interacts via all four forces; in all contexts but as the single down quark in a proton, unstable decaying into an up quark and an electron and electron-antineutrino; the lifetime varies depending on the context. Related to the up quark in the same manner as the electron-neutrino is to the electron.
Charm: Mass between 1150 and 1350 MeV, electrically charged +2/3, spin 1/2 (fermion), interacts via all four forces; can decay into a down, an up, or a strange, with differing decay paths and differing lifetimes. Got its name because prior to the discovery of particles that contained it, there was a good correlation between particle mass and lifetime, but particles that contain a charm quark take an unusually long time to decay, thus leading "charmed" lives, so to speak. The first quark discovered after the up and down.
Strange: Mass between 80 and 130MeV, electrically charged -1/3, spin 1/2 (fermion), interacts via all four forces; can decay into either a down or an up, with differing decay paths, and differing lifetimes not only by mode but by context. Got its name because of the "strange" masses of the particles that contain it. Related to the charm quark in the same manner as the down is to the up.
Top/Truth: The most massive quark, about 170GeV (170,000MeV), electrically charged +2/3, spin 1/2 (fermion), interacts via all four forces, can decay into any other flavor with differing lifetime and decay products. Got its mostly-used name of "top" because it is the "up-est" quark; some physicists prefer the name "truth," as it goes along with "charm" and "strangeness."
Bottom/Beauty: Mass about 4100 to 4400 MeV (4.1 to 4.4 GeV), electrically charged -1/3, interacts via all four forces, can decay into any lighter quark with varying lifetime and decay products. Got its name in very much the same manner as the top, being the "down-est" quark; as with the top, some physicists prefer "beauty" to harmonize conceptually with "truth," "charm," and "strangeness."

For each of the quarks and leptons, there is also an antiparticle; thus, anti-muons, anti-strange quarks, and so forth. The anti-particle has reversed charge, for example the anti-muon has a single positive charge, and the anti-strange a charge of +1/3. Richard Feynman believed that antiparticles were "time-reversed" versions of particles, and presented evidence to support this idea that has never been refuted; however, many physicists reject this interpretation of the evidence, although they don't necessarily have anything better to offer. When a particle meets its conjugate (opposite, like the proton and the anti-proton), the electrical charges and most other charges cancel out, and the two particles annihilate; they become photons, generally, although other charge-neutral possibilities occasionally happen too.

The quarks never occur "free" in nature; no one has ever seen a particle with a charge of less than 1, either positive or negative. They always come either in pairs or in trios (there is evidence to suggest that they may also come in very short-lived "penta-quark" configurations with five quarks, but this has not been definitively classified or proven yet). Collectively, every particle made from quarks is called a "baryon," and this is the great division in the fermions (particles with half-integer spin, like 1/2 or 1-1/2 or 2-1/2), between the leptons and the baryons. The baryons are further subdivided into the two-quark mesons and the three-quark hadrons (and possibly five-quark pentaquark configurations). The mesons are all unstable, but the hadrons have the stable proton and semi-stable neutron as their lightest members.

There are also particles with integer spins (like 1 or 5) called "bosons." These are the force particles, as opposed to the matter particles, the fermions. Since there are four forces, there are four bosons for them:

Bosons
Photon: the boson of the electromagnetic force, massless (mass = 0), spin 1. Stable, and is its own antiparticle. The photon's quantum theory is called "Quantum Electro-Dynamics," or "QED" for short.
Weak boson: the boson of the weak force, mass about 80.4GeV for the charged and 91.2GeV for the uncharged version, spin 1. Unstable with lifetime of about 3x10^-25 seconds, decay into a lepton and its antineutrino (or an antilepton and its neutrino, for the positive charged version, which is the antiparticle of the negative charged version), or various other possible particles. Participates in the decay of quarks into lighter quarks, as well as other possible decays among the leptons. The charged version is the W, with the W- being the particle and the W+ being the antiparticle; the Z is the uncharged or "neutral current" version, and is its own antiparticle as with the photon. The weak boson's quantum theory is combined with QED, with QED forming an offshoot of it; this theory has no cute name, but is generally referred to as either "electro-weak theory" or "Weinberg-Salaam theory," after its creators.
Gluon: The boson of the strong force, mass a few MeV at most (limited by theoretical and experimental considerations, but never directly measured), electrically uncharged, spin 1. The quantum theory of the strong force, QCD, says that there are eight colored gluons. These gluons, like the weak bosons and unlike the photon, can directly participate in the force they mediate. The description of QCD is extremely complicated, and I will shorten it greatly by merely noting that the electromagnetic force has one flavor, the weak force has two flavors, and the strong force has three. The flavors of the strong force are confined, so only neutrally charged particles occur in nature; this confinement is believed to be what makes the quarks appear in pairs or trios rather than singly. Only the baryons participate in the strong force.
Graviton: The boson of the gravity force, this particle is extremely elusive and has never been observed. Its mass is believed to be exactly 0, and its spin to be 2, which accounts for the fact that it only mediates an attractive force, not a repulsive one. Some physicists state that in the absence of persuasive evidence of its existence, it may not in fact actually exist. There is no quantum theory of gravity, although considerable theoretical work has been done trying to devise one.

OK, now that you have the background, the description of radioactive decay is quite simple: when protons and neutrons combine into a nucleus, if you get just the right number, the protons make the neutrons stable, so they don't decay. All atoms, with the single exception of the single-proton isotope of hydrogen, the most common form, technically called "protium," have both neutrons and protons in their nuclei. The reasons this stability occurs have to do with details of QCD. But if the number of neutrons isn't quite right, then the nucleus will be unstable, and will decay into a more stable nucleus and keep on doing that until a stable one is reached. This generally happens via the decay of neutrons, which for historical reasons is generally called "beta decay;" those same historical reasons have the electron that is emitted when the neutron turns to a proton called the "beta particle." Some very heavy elements go beyond this and emit entire helium nuclei, two protons and two neutrons, which for historical reasons are called "alpha particles," and thus this decay mode is called "alpha decay." Only the heaviest unstable nuclei do this; it is a radical move. In either case, alpha or beta decay, the nucleus is generally left in a high-energy state because of the QCD interactions involved. In order to get rid of this extra energy, the nucleus will emit one or more photons; since the energy is quite large, these are generally very high-frequency photons, which for historical reasons are called "gamma rays."

Alpha decay is the result of strong force imbalances within the nucleus, which are balanced by emitting the alpha particle. Precisely when this will happen depends upon exactly how the nucleus is vibrating, or more properly, the protons and neutrons in it are moving around under the influence of the strong force; and since we cannot tell this, we in turn cannot tell when a particular nucleus will decay via the emission of an alpha particle. However, because these vibrations can occur only in very specific ways, there is a certain probability at all times that a heavy unstable nucleus will decay, and thus if we have a collection of many billions of such nuclei, a certain number will decay each second.

Beta decay is the weak force causing neutron decay. Neutrons also vibrate, or more properly the quarks within them do, and as with nuclei, we don't know what the vibration state of any given neutron is, but again as with nuclei there is a certain probability, and given many billions of them, a certain number will decay each second.

Thus, although we don't know when any particular nucleus is going to decay, we do know that a certain percentage will decay by emission of an alpha particle, and another certain percentage will decay by emission of a beta particle, each and every second. And when they do decay, we also know that they will emit gamma rays, and again, so many per second. So although we know nothing of what individual nuclei will do, collectively, we know just how much radiation a particular sized sample of a particular radioactive element will emit.

Now, having described how and why atomic nuclei decay, let me re-iterate that alpha radiation is helium nuclei, beta radiation is electrons, and gamma radiation is photons. It's important to understand that all electromagnetic radiation is photons; radio waves, microwaves, infrared heat radiation, visible light, ultraviolet that gives you sunburn, X-rays, and gamma rays are all the same kind of thing, just at different energies. But alpha and beta are something different from any of these, because they are made from tiny bits of matter, instead of massless photons.

Gamma is the most penetrating radiation; a good thickness of lead is required to stop it. Beta particles are very light, and so they penetrate farther than alpha; but because they're just electrons, which are around us all the time, they generally don't do much damage. Alpha particles are massive, so they do a lot of damage; but they aren't going very fast, being so heavy, so they generally can't make it through a piece of paper, and don't penetrate very far past our skin. You can get a nasty burn from them, but most likely it will just be on the surface, unless you inhale or ingest radioactive material (this is called "radiation poisoning;" "radiation sickness" is the symptoms of the burns you get inside your body from these radiations); gamma burns, on the other hand, are likely to be inside your body, and very dangerous as a result.

When nuclei break down, they actually turn into a different chemical. The number of protons in a nucleus determines how many electrons that nucleus can hold; and the number of electrons in an atom determines its chemical properties, for example whether it's a metal, a gas, or something else, and whether acid will dissolve it or not, and so forth. Thus, when plutonium or uranium or another radioactive element are "spent," they aren't plutonium or uranium any more. They've turned into something else. And in general, that "something else" is lead.

That ought to give you a pretty good idea about how nuclear physics works, and what things it tells us to expect. Let's see you chew on that a while and see what other questions you might have about how this stuff works.