Logging in on AniDB is handled via cookies, which also makes it possible to remember your login information so that you won't have to log in every time you visit the site. In order to login, you can either visit this explicit page: AniDB.net - Login, or you can use the login form on the left just below the menu. In order to not have to login every time you visit the site, you should check the "remember" checkbox by the login form.
If you have any problems with logging in, check here.
In order to log out, you only need to click the logout button, located below the left-hand side menu in the default stylesheet. This link also works: logout.