To remove a user, you first remove all his files, mailboxes, mail aliases, print jobs, cron and at jobs, and all other references to the user. Then you remove the relevant lines from /etc/passwd and /etc/group (remember to remove the username from all groups it's been added to). It may be a good idea to first disable the account (see below), before you start removing stuff, to prevent the user from using the account while it is being removed.
Remember that users may have files outside their home directory. The find command can find them:
find / -user usernameHowever, note that the above command will take a long time, if you have large disks. If you mount network disks, you need to be careful so that you won't trash the network or the server.
Some Linux distributions come with special commands to do this; look for deluser or userdel. However, it is easy to do it by hand as well, and the commands might not do everything.