Debian is a free operating system, but it is a bit more than that as it also provides lots of packages, most of which are precompiled software. There is also lots of software in Debian, that relates to OSM (JOSM, mapnik, osm2pgsql, ...), and some software that is not (mod_tile, tilemill, ...).
One of my long term goals is for it to be possible to install an OpenStreetMap tileserver on a Debian system with a single command. The user would install a metapackage via the package manager, which then depends on all the components needed (mod_tile, postgresql, apache, ...), and then be asked the relevent questions (what data, ...) by debconf (the Debian Configuration system).
Unfortunately, this is a long term goal, as several not insignificant things need to happen first. The first probably being getting mod_tile packaged for Debian, as I currently understand, this is blocking on iniparser (which is not in Debian) being used in mod_tile.
In the short term however, there is now some hopefully comprehensive documentation on building a tileserver with Debian up on the Debian wiki. It should not have too many bugs, as I ran through the whole guide twice while writing it. Some bits still need doing, for example, this guide only covers jessie (the current testing release), and it does not cover getting a slippy map (with leaflet, openlayers, ...) setup.