* Alpha particles:
Alpha particles have a very short lifespan.... generally the distances traveled are measured in inches (less than 12, typically about 3-4) and the particles are stopped by skin. FROM WIKIPEDIA: "They can be absorbed by tissue paper or the outer layers of human skin (about 40 micrometres, equivalent to a few cells deep) and so are not generally dangerous to life unless the source is ingested or inhaled."

What is an alpha particle?
An alpha particle is basically a helium atom with all the electrons stripped off of it. Because of this, it is very heavy (relatively speaking) and does not travel at high velocities, meaning that if it does penetrate, it's at VERY shallow depths.

* Beta particles:
Beta particles are much longer lived than alpha particles... they also can cause more damage to living tissue. There are two types of beta decay. Both involve free nuclear (proton, neutron) particles decaying into high energy (FAST) electrons or positrons and neutrinos (or antineutrinos, depending on which particle decays). Neutrinos (and antineutrinos) are a nonthreatening byproduct of the decay (in a given second, you have 2 x 1010/cm2/sec passing through you every second of every hour of every day. That's 20,000,000,000 neutrinos per square centimeter per second.) Even being in the same room with a neuclear reactor, though it increases your exposure to neutrinos by 100,000 times, is effectively safe. In short: neutrinos do not react with their surroundings (this is a generalization, but it's so close to fact that it is, to nearly arbitrary precision, accurate).

High speed (relativistic) electrons and positrons (basically an inverted electron can cause problems. The elements that the decay occurs in can literally pull electrons out of your body and into itself (the pull to fill its electron shell is more than your body's cell can pull to hold onto it).
"The loss of an electron can cause problems, including everything from cell death to genetic mutations (leading to cancer), in any living thing." (HSW).
Beta particles can be stopped by plexiglas, or tinfoil. Again, only truly dangerous if ingested or inhaled.

* Gamma rays:
Electromagnetic radiation. Not particles at all. can cause mutation by stripping off electrons from atoms and molecules

Stopped by lead bricks. The most dangerous decay product.

* Neutrons:
"The most important force to neutrons is the strong interaction. This interaction is responsible for the binding of the neutron's three quarks (one up quark, two down quarks) into a single particle. The residual strong force is also responsible for the binding of nuclei: the nuclear force. The nuclear force plays the leading role when neutrons pass through matter. Unlike charged particles or photons, the neutron cannot lose energy by ionizing atoms. Rather, the neutron goes on its way unchecked until it makes a head-on collision with an atomic nucleus. For this reason, neutron radiation is extremely penetrating and dangerous." (wikipedia) Neutrons are unstable when seperated from a nucleus and will eventually decay into a proton... generally after striking some random nucleus.