For radioactive particles that decay to known decay products, half life is the amount of time required for a half of the radioactive particles to decay to the known products.
Uranium 235, for example, has a given half life of 703.8 million years. It it will spontaneously decay to form Thorium. If you have a chunk of U-235 (which, if you do, the authorities are most likely after you), half of the Atoms making up this material will have decayed to form the thorium decay product after the half-life of 703.8 million years. (Actually, the decay interactions of this isotope are more complicated than just a simple fission into thorium - see: Uranium 235 for details )
Half life can be directly measured for radioactive materials, although calculations are obviously required to extrapolate if, like U-235, it has a very long half life. I.e., you measure the amount of decay that occurs in a finite interval of time, then calculate the half life from this measurement.