The 'Download OSM Data' option prompts for bounding coordinates.There is also a short description provided on the Mapperz blog. In general consult the full manual on the QGIS site.
Screenshot with OpenStreetMap data loaded from the map here, also a selected road revealing the OpenStreetMap tags on the right Manuals Go to 'Plugins' menu -> 'Manage Plugins.' and find OpenStreetMap. It is a "core" plugin which comes with QGIS when it's first installed, and just needs to be enabled. This new OSM file will not have the original ID numbers for each item so it will happily load into qgis. There is also a somewhat dirty hack by loading your OSM data first into JOSM, copy and paste the data to a new data layer and then save that layer (whatever you do don't upload it to OSM!).
You can convert OSM data to Shapefiles (or just download pre-generated shapfiles) or convert to various other formats for import into QGIS, or write OSM data into to PostGIS database.
There are many other approaches, all a bit technical. If you are working with any recently added OSM data, you'll need to take a different approach. but raises other wider discussion about 'The future of the OpenStreetMap plugin'. This has appeared recently due to needing 64-bit Identifiers. See 'openstreetmap plugin missbehaviour' and 'My lines and polygons have turned to nodes when exported as OSM XML data and imported into QGIS!'. There's a fairly serious bug at the moment.