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TMS (Tile Map Service) is a protocol for serving maps as tiles i.e. splitting the map up into a pyramid of images at multiple zoom levels.
The osgeo.org Tile Map Service Specification is the documentation for this standard. Full support for the protocol involves supporting alternate spatial referencing systems, so TMS can be regarded as half way between the flexibility of WMS and the rigid simplicity of the Slippy map tilenames used on the main OpenStreetMap tile server, and also used by google maps and many other map providers.
TileCache is one of the most popular server softwares for running a TMS with full compliance.
Public TMS Servers
OSM TMS Servers
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Tile usage policy
Tile usage policy
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The Y coordinate
A common partial implementation of TMS provides something similar to Slippy map tilenames but with one important, and somewhat annoying difference. The Y coordinates are numbered from the south northwards (like latitudes and graphical/bitmap axes) in the TMS standard whereas google tile coordinates have their origin at top left (NW). This is really just an unfortunate historical misalignment.
In JOSM or Potlatch 2 you can use TMS tile sources (''e.g.'', for imagery background) which do the Y coordinates the TMS way, by simply inserting a minus in the tile URL specifier. Merkaartor also allows this.