Lag is a situation that arises when communication between the game client and your character's server is delayed. This can be caused by many things: a slow internet connection, ISP troubles, power outages, highly-populated servers, etc. When you encounter lag all forms of interaction with the game world, such as fighting, looting, and talking to NPCs, will occur erratically or not at all.
Lag typically dissipates after some time. After the lag goes away any damage caused or taken by your character will occur rapidly and other characters and mobs may appear to move very quickly. Because of this, severe lag often leads to a wipe.
A more technical disussion is available at Lag on Wikipedia. For a detailed explanation about lag in World of Warcraft see this blue post.