A mailbox provides the e-mail account with 2GB of space to store e-mails on the 1&1 mail servers.
A forward however, will route the incoming e-mails for the account to whichever e-mail address you specify as the target.
All 1&1 e-mail accounts have the option to configure an account to utilize a mailbox, a forward, or both! In cases where an account utilizes a mailbox and a forward (or multiple forwards), the incoming mail will be stored in the mailbox with copies of the mail sent to the specified forwarded address(es).