Stig is the expert driver that trains special guests for the timed track challenge, drives tested cars for a timed lap of the test track and performs other driving duties on the Top Gear show.
The Stig's identity is not shown on the Top Gear show. It is known that there have been at least two Stigs.
The Stig dressed up in the first two series was Perry McCarthy. A British professional racing driver that did race in F1. After announcing in his book that he was the Stig, the black Stig was killed off at the start of the third season of Top Gear when he went off a cliff in a car stunt.
In the second show of the third series, the new white Stig was introduced to Top Gear. The white Stig is probably a shared entity. Ben Collins and Damon Hill are commonly named drivers associated with the white Stig. It is believed Heikki Kovalainen dressed up as the Stig for the Renault F1 drive.
Update: Ben Collins was shown to be the Stig after a court battle about his autobiography, The Man in the White Suit.
The next Stig's identity is still unknown.
It is not Simpson... that is the brand of his racing helmet!
The original name cast about for the Stig was "The Gimp", but original Stig, Perry McCarthy vetoed that name.