It depends on the game:
Morrowind: Whatever suits the character, you couldn't just join every faction so you had to make choices about which direction to go. So I generally picked the race that suited the tone of my character. Orc barbarians, Khajit thieves, Dark Elf conjurers, Redguard warriors, Argonian assassins, High elf wizards etc.
Oblivion: Generally Breton. The beast races felt too much like furred humans and not like beasts and due to the game's tendency to push you to being a fighter-mage-thief many of the other races didn't feel right. But overall I didn't feel my race or even class choice mattered much in Oblivion. The Breton's magical resist bonus combined with big bonusses to some of the more annoying to level up spell schools (unless chain-casting low magicka spells, which I prefer to avoid) made him the kind of logical choice.
I hope Skyrim will let your race matter a bit more again. But overall I really like all of the races, they all fit some type of character really well. I like how every race a quite distinct personality.