The Torre de las Telecomunicaciones is the tallest building in Montevideo, and in all of Uruguay for that matter. It towers over the rest of city and can be seen from quite a distance. The tower belongs to ANTEL, the state-run national telecommunications company.
From the tower you can see a beautiful overview of the city, port, and river. Every so often they give free tours to the public and let you ride up the elevator to the 20th floor where you can look out the massive windows from to see Montevideo on all sides down below.
The windows often reflect light and, at some points, make it seem like you are walking into an optical illusion. It is well worth the visit if you come visit Uruguay.
In the adjacent building there is the Museo de las Telecomunicaciones y el Auditorio which more or less takes you on a tour of telephones throughout the ages. I wouldn't recommend the museum unless you are telecommunications junkie, but the tower is worthwhile.