Versus is a mode that allows the player to play a versus game against a friend or the computer. There is usually a scoreboard to show who is in the lead with the most wins. Versus mode allows the option of 1 to 5 rounds per match.
Similar modes include Team Battle, which is basically the same as Versus mode but with multiple characters, and Online Mode, which can be used to either have ranked or non-ranked matches with different players around the world.
The option to fight against a computer opponent no longer exists as of Tekken 7.