A Gray code (named after their inventor) is a coding system in which 'neighbour' codes only differ in one bit. In a classic binary code, many transitions from one value to the next involve multi-bit changes; e.g. going from 3 (011) to 4 (100) involves changes in all three bits.
As an example, here is the simplest three-bit Gray code (note that the transition from value 07, counting up and wrapping around to 00, also changes only one bit):
Gray codes are sometimes called 'reflected' codes, since the values in one half of the code table are (except for the high bit) a mirrored reversal of those in the other half (e.g. above, the code for 04 is a reflection of that for 03, 05 for 02, etc).
The multiple bit changing behaviour of a classic binary code can be an issue when one is used, for example, in a state machine: when using a counter to hold the state, if the binary-coded state is fed into combinatorial circuitry, the output bits do not all change at the exact same instant. So (to give a concrete instance), the counter output might very briefly display 7 (111) in the process of going from 3 (011) to 4 (100). The combinatorial circuity might therefore produce glitches on its output during the brief period when a false intermediate outputs are present.
Gray codes have other uses too, though - for example in transmission codings.