Which address book are you concerned with?
If it is the global address book or company address book on the server, usually it takes administrator rights to accomplish this task.
However, if it is your personal address book, you should find an icon on your workspace titled "<Your name> on Local" that is your local database of addresses. Within this database, click on the Business Cards icon in the navigation bar, select the name you want to delete and delete it as you would a message from your inbox.
Hope this helps.