Games that have 2-dimensional gameplay cardinality allow the player to move entities along two orthogonal axis. Movement does not necessarily have to be along both axis simultaneously (as in, diagonal movement does not necessarily have to be possible). Typically, 2-D movement is left/right and up/down
Age of Empires
Age of Empires is a strong example of a 2D game. Although the game appears at first glance to be 3D, the game is relatively simple and uses basic sprites, which can only move along a plane, as opposed to anywhere in space.
In Pac-Man, the player controls a character that must navigate a 2-dimensional maze. The player can only move up/down or left/right at a time.
In Tetris, the player controls the left/right movement of a falling block. In addition to moving the block from left to right, the player can increase the speed at which it falls by pressing down. The player cannot make the block move up.
Jets 'N' Guns
This side-scrolling shooter (like most others) allows up/down and left/right movement, but there is no third dimension with which to work.
The player has the option of moving up, down, left, or right at any given time. These are the only movements the player is allowed to make in this platform game, so it is a good example of 2-D gameplay.
Knights of the Old Republic
Knights of the Old Republic is rendered in 3d, but the game engine is based on Dungeons and Dragons, and has completely 2-dimensional gameplay.
The player moves up, down, and sideways through a two dimensional plane. This game is a "side-scrolling" game and therefore 2-D. This is despite the fact that the gameworld is sometimes rendered in pseudo-3D and that the camera often moves in 3-dimensions when it follows Joe into certain areas.
As a side-scrolling platformer, the Mega Man series allows the player to only move backwards and forwards or up and down. Based on these movement options, the Mega Man series illustrates 2-D gameplay.
Escape Velocity: Nova
They player navigates their spaceship from a topdown perspective of the solar system. They can move in 360° by using the left and right keys to rotate the ship and the up key to accelerate forward.