There are two types of lime used in Thai cooking. There are the regular limes that are common here in NZ, known as "manao" in Thai.
These are used to add sourness to a Thai dish, especially in dishes like Somtam (Papaya Salad). It is the squeezed juice that is added that adds the sourness, though often dishes are served in food stalls and restaurants in Thailand with an extra lime edge on the side for people who want to add extra sourness.
The other lime that is used, and is more common in Thai Cooking is Makrut limes ( sometimes known as Kaffir Limes but there are good reasons not to use that term)
Usually it is only the leaves and rinds of Makrut Limes that are used. The leaves (bai makrut) are used in curries and soups to add a citrus twang and fragrance to the flavours. The leaves are tough so are not normally eaten.
The rind is used in curries and pastes.
We have wonderfully handcrafted lime squeezers, made for extracting lime juice.