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.
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.