It is mostly used for radioactive decay. So when an isotope is unstable in it's nucleus (because of too many neutrons, for example), the atom will undergo a process that alters the nucleus (number of protons and electrons).
Radioactive Carbon is an example, C-14. It has 6 protons and 8 neutrons. Its nucleus is unstable because of the ratio of neutrons to protons. In order to become more stable, one of the neutrons (we are going to say a neutron is composed of a proton+electron....this is what our simplistic neutron is) is going to break apart. The electron that is bound in the neutron comes 'flying out', and this is called a beta particle.
The C-14 atom has just undergone radioactive decay. Now the atom has only 7 neutrons...and 1 extra proton (that it got from the breaking apart neutron). So now the Carbon atom is actually a Nitrogen atom with 7 protons and 7 neutrons.....and it is stable. This is radioactive decay.
It is very hard to determine when 5 atoms change, 10 atoms change...but it is easier to see when half the population changes.
If you have a sample of Carbon-14, it will undergo this radioactive decay, and the time it takes for HALF of the Carbon-14 atoms to change into Nitrogen is called the Half Life of Carbon14.
So, radioactive elements have half lives. It is the time it takes for half of them to decay into a new element.
As an aside, half-life is also sometimes used when talking about things like drugs. If you take Drug X, it will be in your blood, for example..and will slowly (or quickly) start to fall apart in the body. THe time it takes for half the drug to change into something else (usually modified by the body or labile bonds get hydrolized), is the drugs half-life. In this case, it is not radioactive, but rather just falling apart inside the body.