Rushing is the act of a mouse (a rusher) blindly trying to get cheese first, even at the expense of other mice, be it killing them in the process, or breaking the map for anyone lagging behind. While rushing can often result in at least one mouse getting in the hole (making sure someone gets a "first"), it often leaves a trail of dead / trapped mice behind.
Rushing may also result in everyone losing, even all the rushers. This can be due to too many mice rushing at once, rushers not coordinating enough, or rushers slowly dying at different obstacles, with none making past all of them.
Many maps are designed with rushers in mind, either promoting it and making way for them to flourish, or tricking rushers by adding traps in the map that are timed / placed in ways to only kill rushers.