Brush is the tall grassy terrain on a map that provides a champion with a form of stealth. Upon entering the brush, a champion immediately (no fade time) becomes invisible to enemy units outside of that particular patch of brush. Visually, this effect manifests by your champion's avatar taking on a partially transparent quality. Opposing champions in different patches of brush will not be able to see each other even if they are within normal sight range. Brush, however, will never block your view of an allied champion hiding within.
Brush can be used in several ways to create a tactical advantage on the battlefield. First, and pehaps most apparently, one can utilize the brush to set an ambush for an enemy champion, emerging once they have passed to cut off their escape. Brush can also be used to while fleeing, breaking line of sight, and forcing pursuing champions to guess which direction you have continued. It can also be used to dodge passing minion waves, to hide your champion while he channels recall, or to force a ranged champion to close to melee range to continue their attack. Remember to use the brush to your advantage on the Field of Justice!
Note: Vision and Sight Wards are subject to the same limitations surrounding brush as other units. Vision Wards placed outside of brush will not be able to see enemies inside.