Your Mailbox stores rewards from daily logins, guild tasks, purchases, special events and some other reward types in the game.
You can access your mail from the World Map by selecting the mail button on the right side of the screen.
You can view mail one by one by selecting the mail you wish to open and then collect the rewards shown by selecting the green Claim button.
Or you can collect all your rewards at once without viewing the messages by selecting the Claim All button at the bottom of your Mailbox.
All mail has an expiry date - usually it's 1 week after receiving the mail (for things like guild tasks or weekly rewards) but daily Guild Wars Rewards expire in 24 hours. Make sure you collect your mail before it expires!