Creating your account
To register, go to http://www.openstreetmap.org/, and choose sign up in the top right-hand corner. This will take you to the following registration form:
At present, you only really need an account on openstreetmap.org if you're planning to contribute mapping data to the project. Outside the main site and API, only the forums and issue tracker use the same username and password as openstreetmap.org. You don't need to register to download data, export maps, or subscribe to the mailing lists. Conversely, even if you're not planning to do any mapping, there are still good reasons to register at the site, such as the ability to contact and be contacted by other mappers.
OpenStreetMap doesn't allow truly anonymous editing of data. The OSM community decided to disallow this in 2007, so that any contributors could be contacted if necessary. If you're worried about privacy, you can register using a pseudonym, and this will be the only identifying information used for your account...