A character is you!
As has already been covered, tilemaps were developed as a way to display backgrounds using relatively minimal computational resources. However, tilemaps alone are difficult to make a complete game with. A method was needed to overlay small moving objects over the tilemap in order to create many of the genres of games we now take for granted. The solution was found inside the tilemap circuitry: at the heart of a tilemap generator is a circuit that can draw a single tile at a specified screen position with horizontal or vertical flipping. This became the basis of early sprite circuits used in games such as Rally-X and Pac-Man.
These early games were limited to sprites that were the size of a single tile and there was a low limit on how many sprites could be displayed at once (often 8 or 16). Nonetheless, many of the games now considered "classics" were made within these limitations.
 Growth opportunities
Almost as soon as there were sprites there was a desire for bigger sprites. This was pursued in multiple directions by hardware makers. Most hardware went the route of making sprites bigger by allowing a single sprite to be made of multiple tiles. Unlike a conventional tilemap the order of tiles used in the sprite was normally fixed in some pattern, but the effect was the ability to show larger sprites. Sega was among the first to experiment with scaling sprites, where a single image was drawn larger or smaller than it actually was drawn at. This was first used in a relatively primitive form in Turbo. A more sophisticated version powered classics like Space Harrier, OutRun, and Afterburner.