Level in Unreal Tournament 3 (as in most FPS games) is a place consisting of a playable area surrounded by some non-playable area. Its purpose is to expand core gameplay of the particular game (UT3 in our case) via layout, pickups (items) and other gameplay related features - and also to provide illusion of a particular place (real or fantastic) using various aesthetics, visuals and atmosphere (AVA) aspects.
Because of the nature of levels in FPS games - and in hard-core multiplayer FPS games in particular - level is often called a map. Level design (mapping) is the process of the creation of the level. It's also the discipline (and art as well) about this process. Levels are created by level designers (or mappers) using level editor and other programs such as 3D modeling software and 2D graphics software. Level editor for UT3 is Unreal Editor for Unreal Tournament 3 - here also referenced as UnrealEd.