Montevideo is the capital of Uruguay.
It's possible to get a ride to Montevideo from the airport by just talking to some friendly looking people.
You can rent a car or take a taxi to the city from the airport - try not to be ripped off though. The "official taxi" company is trying to extort something like 20 US$ for the ride, which is way too much.
Take bus 137 from Plaza Independencia (or somewhere else), of course, make sure you take it in the right direction (which is confusing in the beginning, because it seems to make a large detour). After half an hour or so you have to get out at the crossroad with Ruta 1. There is a gas station not far from there. Try to aim for black numberplates (which are Argentinian).