Medal play, or strokeplay, is the total number of strokes that each player takes to complete the round.
Net golf medal play is the total number of strokes using handicaps.

Most clubs organise a monthly medal where you play a stroke play competition where all shots count. The prize is a medal engraved with month and year and possibly other incentives.

All major golf tournaments like The Open Championship and The Masters are competed for in medal play.
The Ryder Cup, Walker Cup & Presidents Cup are played as matchplay competitions.