Half-life is a term commonly used in radioactivity. It defines the time taken for a radioactive source to halve its activity. A radioactive source will emit radiation (energy) through the emission of particles or photons. How many of these particles are being released every second (or any arbitrary unit of time) determines a substance's activity. This can be seen mathematically through analysing the radioactive decay equations listed in the AQA formula list here: http://filestore.aqa.org.uk/subjects/AQA-PHYA4-5-INS-JUN12.PDF. Furthermore, the maths used to derive the equation for half-life, can be explained through these equations; I would be happy to show you if you wish. In conclusion, half life is how long it takes for a radioactive substance to become half as radioactive as it initially was.