Anything presidential has something to do with a president, or with the office of the presidency. A line of black cars with flashing lights driving down a Washington, DC street is probably the presidential motorcade.
In the US, the presidential residence is the White House, and the presidential term is four years. A presidential election can seem like it lasts even longer. You can also use this adjective for things that are typical or appropriate for a president. A presidential wave, whether the president gives it or not, is commanding and confident. And it's well known that candidates who don't look presidential aren't often elected, even if they're more than capable of running the country.