Web map applications generally make use of imagery “tiles”. Image tiles are very large pictures of the world at a certain zoom level that have been chopped up into identically-sized pieces (usually 256x256 px). As you pan around a map, the map application translates the lat/lon coordinates into corresponding X/Y integer values at the given zoom level, and uses those to request a tile from a URL on the server. The server generally has millions of individual image tiles on disk, and can easily return the right file with no special work needed (assuming the tiles have been stored on the server in a known folder structure).
Cesium can make use of many common imagery tile servers and structures, and allows you to specify what URL structure is needed for a given server when you set up a Cesium ImageryProvider. In addition, Cesium also needs terrain data to show 3D terrain. Cesium has defined a pair of terrain data file formats that it can use, and also expects those to be available on a server in the same fashion as imagery tiles.
So, the terms "terrain server’, “imagery server”, and “tile server” all refer to some web server that serves up tiled data for imagery, terrain, or both.