Choosing an API/library
Getting started with OpenLayers – a mature, feature-rich library
Getting started with Leaflet – a new, lighter library
Choosing a tile provider
Apart from very limited testing purposes, you should not use the tiles supplied by OpenStreetMap.org itself. OpenStreetMap is a volunteer-run non-profit body and cannot supply tiles for large-scale commercial use. Rather, you should use a third party provider that makes tiles from OSM data, or generate your own.
Paid-for providers: see list.
Or go on to find out how to generate and serve your own tiles.