Thick means "powerful".
There are thick stones, thick plays, thick shapes, thick styles of play, thick territories. A player's whole board position may be thick overall.
It is better to be thick than thin. It is better to be thick than heavy. Learning to distinguish thick play from heavy play is very important to one's go progress.
A player with a thick overall position typically has an advantage in terms of fighting, forcing plays, and ko threats. This overall thickness enables her to play enterprisingly. Even if fights or ko fights do not develop, the threat that they will do so can be effective.