The characteristic of a thief is that he goes where he wishes, and you can't quite predict where he'll be discovered (except by total physical isolation). Thieves have a metal dagger (typically copper or silver), and little or no armor.
When a thief is discovered, it is because he has moved next to one of your units (dwarf, animal, or friendly merchant). At that point, the game will pause, and you'll get an announcement. You can zoom to the location if you wish.
As soon as you unpause, the thief and the units near it will act according to their nature. There may be combat; more typically, civilian units will run in some random direction away from the thief, and the thief will run toward one of the map edges.
Earlier today, some human merchants were visiting my fort, and one of them discovered a kobold thief, just inside my walls. The merchant took a swing at the thief, and missed; the thief counterattacked, and stabbed his dagger straight into the merchant's brain, killing him instantly. The merchant's guard ran straight off the map; the other merchants, some of whom were already inside my trade depot, packed up and left.
But that's an atypical outcome. More often, one of these things happens:
(1) One of your civilians stumbles upon the thief. The thief slashes the civilian unit in the foot, sending her to the hospital, and then runs off.
(2) One of your military units stumbles upon the thief, and kills him, either in one attack, or by wounding him so that the dwarf and his squadmates can dogpile the thief.
(3) One of your military units stumbles upon the thief, but fails to kill him. The thief runs, and the military unit chases him. Unless the military unit has ranged weapons, the thief will probably escape, because the military unit can't outrun him.
Also remember that kobold thieves avoid all traps. Goblin snatchers do not.