Go
Go is a 3000 years old game, originally from China. It is played on a rectangular grid of size 19 × 19, and the players play by alternatingly placing pieces of their colour on the intersections on the board. The object of the game is to get the largest territory.
There are many similarities between Go and Hex, and people who like Go often find Hex interesting, and vice versa.
Some basic information on Go can be found in the Wikipedia article on Go. Complete rules are available at Sensei's Library, which is a wiki for Go.