"Turn" is problematic since it means different things in different traditions. For example, in standard chess terminology, one move consists of a turn by each player; a ply in chess is a half-move. Thus, after 20 moves of a chess game, 40 ply have been completed, 20 by white and 20 by black. In the game of go, by contrast, a ply is the normal unit of counting moves; so for example to say that a game is 250 moves long is to imply 250 ply.
In computing, the concept of ply is important because one ply corresponds to one level of the game tree. The Deep Blue chess computer which defeated Kasparov in 1997 would typically search to a depth of between six and twelve ply to a maximum of forty ply in some situations.