Pumping is the phenomenon that occurs when the compression of an audio signal becomes audible. The primary cause (typically) is the attack, or transient response of the compressor. Say when you go from no signal, to some audio sound the compressor will reduce the gain based on the set compression ratio. If it takes long enough for the compressor to reduce the level to it's final level (assuming a constant audio signal) that it is detectable by the human ear, then it is said to be "pumping". One way that this is most noticable as well is if there is a fairly noisy signal chain and the noise level can be heard to drop when the audio starts. When the audio goes away, the noise comes back up. This is sometimes referred to as "breathing", or the more technical term "noise modulation".
Pumping and breathing are caused by cheap compressors, poorly adjusted compressors and weird guitar guys who set their compressors for "effect".