A share is a hash smaller than the target for difficulty of 1* (see clarification at end). Every hash created has a 1 in ~4 billion (2^32) chance of being a valid share. In comparison if the difficulty of network is 2,000,000 then a share is 2 million times "easier" to find than a valid hash for the block, and on average it will take 2 million shares (8 quadrillion hashes) to find a valid hash for the block.
Shares have no actual "value". To the bitcoin network they are worthless. The only hash with any value is the one that is smaller than the target (determined by difficulty).
Since a miner has no control over when he will generate a share (they occur randomly roughly once per 2^32 hashes) he can't cheat the pool. When one miner in the pool finds the solution hash (the only hash worth anything) rewards can then be split by number of shares submitted. There are multiple reward methods (Proportional, PPLNS, SMPSS, PPS, etc) that calculate the "fair share" of the reward but all use a share as a proxy for work completed by the miner.
It is important to emphasize that a share has no actual value. The only hash with value is the one that solves a block. A share is merely an accounting method to keep the miners honest and fairly divide any rewards earned by the pool.
There is no need to keep track of shares in solo mining because you will not split the reward and can't cheat yourself.
Usually pools use a difficulty of 1 as the target for a share. Technically any difficulty (less than network difficulty) could be used. Each miner's share of the reward would remain the same regardless of what difficulty is used.
