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. Registering with openstreetmap.org requires a valid e-mail address, but this is never disclosed to any other user under any circumstance, unless you choose to do so.
It is possible to change your display name after registration, and this changes it for all current OpenStreetMap data. However, it won't change in any archived data, such as old planet files.
Once you've completed the registration form, you'll receive an e-mail asking you to confirm the registration. Your account won't be active until you click on the link in this e-mail. Once you've activated your account, you can change your settings, as follows:
You can add a short description of yourself if you like, and add a photo of yourself or some other avatar. You can also set your home location by clicking on it in the small slippy map on your settings page. This allows other mappers nearby to see who else is contributing in their area, and allows you to see them. You don't have to use your house or office as your home location; any place that gives a good idea of where you'll be mapping is enough. Adding a location may lead to you being invited to OpenStreetMap-related events in your area, such as mapping parties or social events. If you do add a location, you get a home link in your user navigation on the home page that will take the map view back to that place. You'll also see a map on your user page showing other nearby mappers limited to the nearest 10 users within 50km.
If you know other mappers personally, you can indicate this by adding them as your friend on openstreetmap.org. This is just a convenience to you, and your friends aren't publicly shown on your user page, although anyone you add as a friend will receive an e-mail telling them you've done it.
Once you've completed the account settings, you can view your user page (shown in the following screenshot). You can do this at any time by clicking on your display name in the top right-hand corner. This shows the information about yourself that you've just entered, links to your diary and to add a new diary entry, a list of your edits to OpenStreetMap, your GPS traces, and to your settings. These will be useful once you've done some mapping, and when you need to refer to others' activities on the site.
Every user on openstreetmap.org has a diary that they can use to keep the community informed of what they've been up to. Each diary entry can have a location attached, so you can see where people have been mapping. There's an RSS feed for each diary, and a combined feed for all diary entries. You can find any mapper's diary using the link on their user page, and you can comment on other mappers' diary entries, and they'll get an e-mail notification when you do.