It's hard to answer your post without knowing what OS you're running. I will assume 10.3 and try to answer.
1) Log in as the old admin user
2) Open System Preferences
3) Click "Accounts" icon
4) If the "lock" at the bottom left is locked, click it to unlock it. Enter your password when prompted.
5) Select the new user who you wish to be an admin
6) Click the "Security" tab
7) Check the "Allow user to administer this computer" box
8) Log out
9) Log in as new admin user
10) Delete old admin user in System Preferences->Accounts using the "minus" button in the left pane.