Assault is a custom game type played by two teams, where teams must try to fulfill a certain mission. In Double Assault both teams have an objective to fulfill, in Classic Assault, one team must try to prevent the other team from reaching their objective.
The level is usually a large tunnel-like one, with doors to be opened and trigger crates to buttstomp. One team is attacks, and it is their mission to open all doors, triggers, and overcome all obstacles, in order to get to the defending team’s base or the final target, usually a trigger crate or destruct scenery, which must be destroyed. The other team must prevent the attacking team from reaching their target by using whatever means necessary (although brute force is often used, rather than diplomacy). For most assault levels, the defending team can only win if they keep the attacking team out of their base for a given amount of time (which is set before the game starts). The attacking team can only win by destroying their target before the time limit is up.
The first pubicly released assault level was assault3 by Syntax. It was intended to be part of a 3-level pack by several people, hence the name, but Syntax’s level was the only one to be released. The concept was inspired by the Assault mode in the Unreal Tournament game.
(Copied from <A href=http://www.jazz2online.com/J2Ov2/wiki/?assault>here</a>. There's some slight inaccuracies but no major ones)