A delegated mailbox is a secondary mailbox that is distinct from your own mailbox, however, you have been granted access to view and manage it.

Log in to your Gmail account. (If your Gmail is already open and the (delegated) account was recently added to your account, you will need to refresh your browser window to see the new access. At times, this may take at least 30 minutes.)
In the upper-right corner of the page, click the drop-down arrow next to your email address, and select one of the listed (delegated) account.
Select the shared mailbox from the list that displays. Your account will be listed as the default and your delegated mailbox(es) as (delegated).
A new tab will open displaying the selected delegated mailbox.

Browser not supported