What is radioactivity? How does it relate to nuclear radiation? How does it relate to the nucleus of the atom?

The nucleus of an atom is the site of two of the most
powerful forces in nature, and they are acting in opposition. The protons of the atom
all have a positive charge, and they are being pushed apart by electromagnetic
repulsion. Nuclear binding energy opposes this, holding the nucleus together. Because of
the delicate balance between these forces, there are some configurations of the nucleus
that are unstable, and they undergo various forms of modification to achieve
stability.

In some cases a particle consisting of two
protons and two neutrons is ejected from the atom. This is called an alpha particle, and
the emission is referred to as alpha radiation.

In beta
radiation, the reorganization of the nucleus is a little different. In this case a
neutron is split, creating an electron (which is released and is called a beta particle)
and a proton (which remains in the nucleus).

Gamma
radiation is not comprised of particles, it consists solely of high-energy
electromagnetic waves. The emission of this energy allows the nucleus to settle into a
more stable configuration without emitting any
particles.

The main danger that radiation presents to us is
the potential for damage to our cells, and especially to our DNA. Although alpha
particles are fairly low speed and consequently easily stopped, the higher energies of 
beta and gamma radiation allow them to penetrate into our cells and do damage on the way
by. Imagine being shot by a number of very tiny bullets, and that will give you some
idea of what radiation does to the body.