You can use any username. Note that it is public and all your edits are associated with it. Your e-mail address will not be shared, but other users can send you private messages that are forwarded to your e-mail address. Your password must be at least 8 characters long.
You should receive a confirmation e-mail with a link - click this link. You can now select your hometown by clicking on the map, so that others that can see who is active in the area.
Browse to http://www.openstreetmap.org/ and choose any part of the map (for example, an area that has a mistake) and click on the Edit button (above).
The iD is currently the default editor on OpenStreetMap. When you launch it for the first time you can follow a walkthrough that will give you a quick tutorial on how to edit the map. You can also chose to skip the tutorial, but we highly recommend trying it out.
Of course that was only a small start and there are many other ways to work with the OSM database. Offline editors(JOSM, Merkaartor) offer more features than the iD browser, but must first be installed.