OpenStreetMap is the free wiki world map, an open volunteer-driven initiative to collaboratively create a map of the world, and release the map data under a free and open license.
Some developers may be looking for data downloads. You can download the entire planet's worth of map data from http://planet.openstreetmap.org and keep in syc with community map updates, using the "diff" downloads. Or you can download smaller more manageable country-sized, or city-size "extracts" of OpenStreetMap data.
We do have a thing called the "OpenStreetMap API". This provides read and write operations on the raw map data of the OpenStreetMap database. It is actually primarily for OpenStreetMap editing software, and although any developers can create a new editor tool, this is not a small undertaking, and should be carried out with careful consultation with the OpenStreetMap community. In other words... this thing called the "OpenStreetMap API" is probably not what you're looking for!
You can use the OpenStreetMap API to download small chunks of map data, but for this purpose it's also worth checking out some other options. In particular the OverpassAPI http://wiki.openstreetmap.org/wiki/Overpass_API offers some flexible querying by tags etc, which may be useful for your app.
Whether you're wrangling raw geo-data or presenting a web-map mashup, the possibilities provided by OpenStreetMap, and by third parties within the OpenStreetMap ecosystem, offer endless potential for geo map hacking fun! If you need help with development, join the osm-dev mailing list, or other OpenStreetMap contact channels: http://wiki.openstreetmap.org/wiki/Contact