In Active Directory Users and Computers go into the users properties that has the mailbox that you want to grant access to. This should be on a server or pc that also has the Exhange 2003 tools installed.
Click on the Exchange Advanced Tab, and click the Mailbox Rights button.
Click the Add button and choose the user who you want to be able to access the mailbox, and click OK.
Back on the permissions page highlight the user you just added and below tick the Allow column for 'Full mailbox access'.
Click Apply and Ok.
Back on outlook of the person who needs to be able to access the mailbox, goto the Tools menu then Account settings.
On the E-mail tab highlight the Microsoft Exchange account and click the Change button above.
Click the More settings button, and then the Advancded tab.
Click on the Add button and type the name of the users mailbox to be accessed and click OK.
More than one mailbox can be added here. If so repeat the previous steps.
Click OK, then Cancel, and Close.
You will be prompted to restart Outlook.