Determining whether a meteorite comes from Mars is a matter of analytical chemistry. There is gas trapped in glassy inclusions in the rock, and you measure the chemical composition of that gas. The Viking lander measured the composition of the atmosphere of Mars when it arrived there in 1976; if the composition of the gas trapped in the meteorite matches closely the composition of the Martian atmosphere, then the meteorite almost certainly came from Mars.

There are about a dozen meteorites (known as "Shergotty-Nakhla-Chassigny meteorites") whose trapped gases match the known composition of the Martian atmosphere, down to trace gases whose concentration is only a few parts per billion. ALH84001 was determined to be one of these meteorites by D.W. Mittlefehldt in 1993 (and this set off a flurry of interest in it, culminating in the announcement you are reading about now).

Click here for more information (from NASA Johnson Space Center).