The first one is the official OpenStreetMap data, which contains the "planet file" - i.e. all the data for the entire world. But because OSM has so much stuff in it, the planet file is a whopping 76 GB, which can take a long time to process for most tasks. I also recommend using the torrent file for faster download speeds.
As a result of the planet's size, the German company Geofabrik provides unofficial "extracts" of the data, which are filtered down to a specific region. E.g. all the data in a particular continent, country, or U.S. state. If you click on the "Sub Region" link it will show countries, and if you click on those it will show states.