Tile Drawer

Tile Drawer is a free, quick-start, out-of-the-box renderer and server using the standard OSM toolchain for Amazon EC2, other virtual servers, and fresh installs of Ubuntu on any hardware. Choose a geographic area, select a map style, and Tile Drawer will upgrade your stock operating system to an working, running tile server.

(Amazon EC2 is fairly often used for tile hosting but has some caveats: see explanation.)

TileMill and MapBox

TileMill is a pre-built application containing the Mapnik rendering engine, a code editor, and a data inspector. It is available for Linux and Mac OS X. It is built by Development Seed who also offer MapBox, a paid-for hosting service for tiles created with the application.

Third-party providers

Several OpenStreetMap service providers will be happy to work with you to host the map tiles with the stylesheet you have designed. See the provider listing.