The term Gameplay Mode often refers to one of three core philosophies.
The vast majority of games published will have some form of single player gameplay. Single-player gaming often presents itself as a campaign, usually with a story that crosses over increments within that campaign.
Multi-player functionality is games is often offered alongside a single-player mode, in some cases being a more prominent part of the game's appeal than the single-player mode; however, there do exist games who's sole purpose is to play multi-player, such as Twisted Metal Black: Online.
While this, strictly speaking, is technically multi-player, it is such a vastly different gameplay experience, that it would be remiss not to address it directly. Massively Multiplayer is the direct ability to interact with a vast amount of players to achieve a common goal, in comparison to the limited number of players that would be experienced in a typical multi-player setting.