The fruit of a citrus tree native to Southeast Asia, Citrus hystrix, Citrus papedia. The leaves and the peel of the fruit are key ingredients for Thai cookery. The tree itself is smaller than most citrus at about 10ft (mine is only a baby, 3ft), it has nasty thorns and distinctive "double leaves" where at the end of the regular leaf, it grows a second heart-shaped 'leaf' that is not actually a leaf at all, but an enlarged petiole. The lime itself is equally distinctive, smaller than a regular lime, deep green in colour (when over ripe it becomes a pale gold green) and covered in a deep wrinkled texture.
The skin of the lime is grated and added to thai curry pastes, the juice however, is curiously not extensively used in cookery, but is diluted and used as a rinse said to keep your hair shiny. It is the leaves where most of the action is (from a culinary viewpoint). Used whole, they add an unmistakable flavour and intense perfume to curries and soups such as tom kha gai. Cut into optic fibre width chiffonade, they are scattered over dishes to add a lively garnish and mixed through salads.
If you are trying to buy some leaves or fruit from an ethnic market, it may help to have some regional names. In Thailand it is called makrut, limau purut in Malaysia and jeruk perut in Indonesia. If you can find a tree at your local nursery, they are fairly easy to tend and makes an attractive potted plant. Be warned however, if you don't live in a tropical or semi-tropical climate, don't expect too much fruit. I live in temperate Sydney, Australia and have only grown 3 (prized) fruit in 5 years.