You can get a Planet extract. See http://wiki.openstreetmap.org/wiki/Planet.osm#Downloading
The best option is probably Geofabrik, they provide extracts for most countries / parts of Europe. Though they don’t actually have an extract for the UK, but they do have them for “Great Britain” and “Ireland”, so you could combine these to get all of the UK.

The usual software for rendering OSM maps is Mapnik. It will generate the map tiles which you can use in your application. http://wiki.openstreetmap.org/wiki/Mapnik
For making a slippy map, you can use OpenLayers, a JavaScript library. This lets you add markers and interact with the map etc. http://wiki.openstreetmap.org/wiki/OpenLayers
There are a few other JavaScript ways of showing an OSM map, eg Cloudmade API or Google Maps APi. Some of them are listed here: http://wiki.openstreetmap.org/wiki/Deploying_your_own_Slippy_Map