If you want to play sport there are, as you would expect, several recreation centres and swimming pools in the city. This site (in Slovak, but with an easy-to-navigate map with symbols) is a good place to find one, and check prices and opening hours.
If you want to watch, football and ice hockey are the main winter sports. Bratislava will co-host the 2011 Men's Ice Hockey World Championships (with Košice, in eastern Slovakia) and the main stadium is currently being rebuilt. Tennis is popular in the summer. There is a horse-racing track in the Petržalka district.
Mode No graphics is currently switched on.