The mailbox is where the characters of Stupid Mario Brothers get their mail. Mario and Luigi's address is 1234 Rainbow Road, Somewhere Dumb, CA, 45678, Republic of the Democratic (this could be wrong seeing as it is from Bowser, but it is in Mario's dream).
In Season One, almost every episode started with Mario and Luigi coming here and receiving their mail (usually from Bowser).
At the end of Operation: Blind Storm, Mario and Luigi check their mail finding a letter from their landlord, whom is evicting them because of their fail to pay rent (because they used gold coins which is not real money).
The mailbox returned in Season Four. In the trailer, Mario is seen getting a letter when Wario approaches him, saying that this time, it is game over for him, but Mario replies, "As if." The two then have a fight using their powers, but it turns out to be a dream.
Ganon later sends a letter warning that he is coming to the Real World, but Mario and Luigi ignore it, but as it turns out, Ganon's letter was serious and he is coming.
The mailbox later appeared in Episode 62 where Donkey Kong had visited it to take Mario and Luigi's mail. He had however failed in doing this and claimed there was no mail, making the brothers hire UPS to give them their mail.