Apparently, the Babylonians (it might have been another group in that area/around that time) used base 12, derived from the number of knuckle bones of the four fingers on one hand. The thinking is that it was easier for the sheep-herders to count their flock - more counts per hand.
Base 60 was also used, but it is an extension of base 12. 60 has a lot of integer divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30), which was useful in a pre-floating-point world. Another extension, to 360, made dividing circles easier for survey and astronomy purposes. We still use remnants of these systems: 60 seconds = 1 minute; 60 minutes = 1 hour; 360 degrees equals one full turn, demonstating the utility of the bases.
During the initial development of the metric system, the French tried to rationalize the circle, by dividing a right angle into 100 "grads" (360° = 400 grads) instead of 90°, but aside from a few specialized fields (surveyors, I think, may use the grad, but I don't know of anyone else), grads didn't catch on.
I think the resistance to grads was largely because of the British Admiralty's development of circumnavigation techniques. The earth turns 360° in 24 hours, thus an hour is 15 degrees (360/24). When you are determining longitude by comparing local noon (measured with a sextant) against an accurate clock (chronometer) set to Greenwich Mean Time (GMT), the arithmetic was simpler with degrees than grads. Remember, we are talking of a period from the early 1700's to the mid 1900's - there were no calculators, and definitely no GPS. Navies and air forces still teach navigation by these techniques - it is "Plan B" for when the "electrickery" goes south on you.
SJQ