From the Super Mario Wiki
Mr. President (or just The President) was the President of the United States during many Mario cartoons. He has made non-speaking appearances in a few Mario cartoon episodes and was also featured in a Club Nintendo comic. Though never called as such, The President from the cartoon episodes appears to be a caricature of George H. W. Bush, who was the President of the United States around the time that the Mario cartoons originally aired.
The Super Mario Bros. Super Show!
In The Super Mario Bros. Super Show! episode "Plummers Academy" and its live-action segment "Glasnuts", Mr. President has a small role in both the animated and live action segments. In "Plummers Academy", Mr. President was giving Soviet President Mikhail Gorbachev a drive-by tour around Brooklyn, when the street started flooding with water. Mario and Luigi, determined to prove themselves as great plumbers, dived down into the sewers to stop the source of the flood. Later on at the White House, Mr. President awarded Mario with a Golden Plumber's Helper, and Luigi with a Golden Wrench.
In "Glasnuts", Mr. President simply calls the Mario bros. to tell them that Mikhail Gorbachev is coming over.
In the live action segment "Caught in a Draft", Sgt. Slaughter visits the Mario Bros. to train them for the army, as ordered by the President. Later on, Mr. President calls, but Sgt. Slaughter answers the phone, and learns the truth - he was supposed to recruit the Ratigator, not the Mario's.
In the live action segment "Captain Lou Is Missing", the Marios form a search party with Cyndi Lauper to help locate Lou Albano. Towards the end, Cyndi receives a letter from Mr. President, who decrees that Lou Albano is officially missing forever.
The Adventures of Super Mario Bros. 3
In The Adventures of Super Mario Bros. 3 episode "Reptiles in the Rose Garden", despite the fact that the White House is lifted up from the ground and warped to Dark Land, Mr. President never notices, as he spends the entire episode talking on the phone. Even when King Koopa sinks the White House to the bottom of Mushroom Sea does Mr. President not notice what's going on. Instead, his wife does all the talking when Mario arrives to save them.
Eventually, Mario returned the White House to Washington, D.C., and Mr. President was able to continue his job with running America until the end of his term.
Donkey Kong in: Banana Day 24
The President also made an appearance in the Club Nintendo comic "Donkey Kong in: Banana Day 24". In the Kongs' jungle, it is snowing, a phenomenon never seen before, and from the TV news the Kongs learn that the President has invited for a summit to search for a solution to the problem of suddenly dropped temperature all over the world. Donkey Kong, Diddy Kong, Dixie Kong, Cranky Kong and Kiddy Kong decide to travel to the United States and talk to the President.
They meet him in the White House along with German chancellor Helmut Kohl, and after some heming and hawing the politicians explain that the cause of the climate change is a giant UFO pulling the Earth away from the sun. The President agrees to put a space shuttle at the Kongs' disposal so they could fly into space and convince the aliens to release the planet. Their mission succeeds, and so the Kongs return to Earth, being hailed by the people. The President and Kohl use this chance to showcase themselves with the saviors of mankind.
In the comic, the President appears as a likable and helpful person, although not lacking some traits considered typical of politicians, such as refusing to come out with the truth and enjoying to be in the limelight. He is blond-haired and is wearing a brown suit.