We shall start with the basic one. Radiation can be classified into 2 different categories, namely ionizing radiation, and non-ionizing radiation. For the latter, the examples are radio wave, microwave, visible light, infrared and so on. Now, the nuclear radiation fall in the first category, ionizing radiation.

Well, simply say, ionizing radiation is radiation that cause the atoms to ionize. It means that we have one positive atom and a few free electrons. In chemical insight, this ionization cause the molecule to break apart, since chemical bonds are base on delicate equilibrium between no. of electron and no. of proton.

This ionization, on molecular level, cause 2 main consequences, which are degradation and cross linking. In a degration process, let say in a long polymer, the chain would be cut into pieces (degrade). In cross linking, the radiation would give the monomers a chance to glue to each other, forming a longer chain.

Seem so far, but these events are actually closely related to us. The degration that i just talked about can happen in our DNA (it means our DNAs break apart). The broken strand could be match by other broken part, causing mutation

Now let look into our cells. what can possibly radiation do to our cells? As I told u, radiation can kills our cells and cause mutation, perhaps to our child later on. If our cells start to die, so do the entire organ, if our body cannot repair the damage. Normally our body can repair radiation damage, but we cannot tolerate too much damages at one time.

Now on much general points of view, we can summarized the effects as
a) instant death (if we received high dose)
b) cancer induced by the radiation – we still argue whether low dose can induce cancer, but we are certain that high dose can induce cancer at some stage in our life. Note that not all scientists believed in the word induce, but they would rather say radiation increases the possibilities of getting cancer.
c) physical symptoms such as hair lose, nausea, pain in stomach, skin reddening etc.
d) impair growth on children
e) mutation, to later generation.
f) memory loss, not widely reported

Two major incidents involving radiation are the detonation of atomic bombs in Hiroshima & Nagasaki, and the Chernobly reactor incident.
References :
Too complex? Let me know your academic background so that i can change it to much simpler words. Or did u just want general effects of radiation? Maybe u should clarify further.