How can you tell if something will be a good radiation blocker/absorber? Does it have a certain structure or density?
It depends on what you want to stop. Heavy charged particles such as alphas are emitted at a lower velocity and interact primarily through coulomb repulsion. So, as soon as the particle leaves vacuum, it interacts with many electrons simultaneously, causing excitation or ionization of the electrons and leading to a quick energy/velocity loss of the alpha. It turns out that up to 1/500 of the energy per nucleon is lost with each interaction with an electron. So it will quickly stop in anything.
Beta particles are a step up in complexity, as you can have electron-electron or electron-nuclear interactions that can affect both the energy and the direction of the beta particle. The primary means of energy loss though is through ion pair production (I think), and you lose ~35 keV for every ion pair produced. The one thing you have to watch out for is bremsstrahlung radiation, which occurs when a beta particle is accelerated in the electric field of a nucleus. So, you want to watch out for your higher Z elements and stick to something with lots of carbon or aluminum to minimize the bremsstrahlung radiation. A good rule of thumb is that the ratio of energy loss by bremsstrahlung to the energy loss by ionization is EZ/800. Where E is the energy of the beta and Z is the atomic number of the shielding. By the time you get to lead, you lose most through bremsstrahlung, so your shielding is not very useful. That is why you always block the beta particles and then the gamma rays if you have a mixed source.
Gammas are even more complex as they can lose energy by several different methods, the most common being photoelectric absorption, Compton scattering and pair production. All of these processes increase by the atomic number of the shielding nucleus. So here you want the heaviest and most dense material that you can find. Lead is normally chosen as it has a relatively high Z, is quite dense and is also quite cheap.
If you want to stop neutrons, then you have another problem as they only indirectly lose energy with interactions with electrons, and most energy loss is through nuclear interactions. Then you have the added problem of needed different methods for different energies.
I think I have most of that right, but no promises