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Gee (pronounced jee) means to turn to the off side (away from the driver). Haw means to turn to the near side (towards the driver). In the U.S. the driver of draft animals stands on their left -- so animals will turn right to the gee command, and left to the haw command. In England the driver stands to the right of the animals, reversing the relative directions they indicate (i.e. an English trained team of horses will "haw" to the right, while an American trained team will "haw" to the left -- in both cases towards their driver.) As James Lloyd Clark points out, "Generally, work horses are not subject to a lot of international travel so the fear of great confusion on the farm is minimal."
The English meanings are used for sledding in Alaska and Canada.
The song Pony Time, recorded in 1961 by Chubby Checker and a number one hit that year, includes the lyrics
Now you turn to the left when I say gee,Boogety, boogety, boogety, boogety shoo.
You turn to the right when I say haw,
Now gee, ya ya little baby,
Now haw, ya oh baby, oh baby, pretty baby,
Do it baby, oh baby, oh baby,