Serving Tiles
Tiles are the individual images, that when used in combination with another tool (e.g. Leaflet, Open Layers, Marble, ...) form a map.
Tileserver Components
Database
Usually postgresql, with postgis
Renderer
Usually Mapnik, and renderd (part of mod_tile (not in Debian, no ITP yet))
Stylesheet
opentreetmap-carto (not in Debian, ITP #744842)
Web Server
Usually Apache with mod_tile (same as above (not in Debian, no ITP yet))