Odd time signatures are simply time signatures with an odd number of beats in a measure. Things like 5/4, 9/4, and 11/8 are all examples. Most Western music is composed with even time signatures, like 4/4 and 6/8, but occasionally that extra beat comes along in music like jazz, progressive rock, Indian classical music, even movie themes like Mission Impossible. It is a fairly uncommon topic in Western music, but has been practiced by Eastern musicians for thousands of years.
Playing in Odd Time
Playing (and even listening to) odd time signatures is something that must be practiced a lot. For <insert your age here> years you have been bombarded with common time by TV commercials, the radio, high-school dances, and anywhere else you hear music. This tends to train you to be able to dance, tap your foot, or bob your head to the seemingly natural rhythm. If you had grown up listening to nothing but 5/8, you would be able to dance like an Armenian, but Britney Spears would probably make you look like you're dancing on mousetraps.
Thus, to become proficient with odd time signatures, you must listen listen listen and count count count count count! The easiest way to count any time signature is to subdivide the measure, which means counting the beats in groups of two and three. For example:
To count 7/8, you can divide the measure into 2+2+3, which is counted 1 2 1 2 1 2 3.
You could also divide it into 2+3+2, which has a completely different feel. Listen to some of the examples below and count along out loud; you will find that it is not so unnatural after all!
Many modern bands have experimented with odd time, others are masters of it. There is even a genre built around odd time signatures, called math rock. Anyone interested in learning more should listen to these bands/songs:
This is not nearly a complete list, but /msg me if you would like to add something!
Remember that odd time signatures are a relatively new concept to us, but they are slowly gaining recognition in the Western world. Therefore, there are still many undiscovered facets of it and plenty of room for creativity. Now off, young grasshopper, and rock those odd times like a three-legged man at a square dance!