He ran against President John Adams
The first case of this was Thomas Jefferson, who ran against John Adams.
Regan?
President Arthur never ran for President.
President George W. Bush ran against Senator John Kerry in 2004. President Bush won. Jim Carey, an actor, never ran for political office and never ran against any President of the USA.
Taft.
James Monroe
Adlai Stephenson
George H. W. Bush ran against Ronald Reagan in the 1980 Republican primaries, but lost. Bush would eventually become Reagan's Vice President.
If you are asking who he ran against, it's Thomas Dewey.
Thomas Jefferson
In 1992 the incumbent president George H. W. Bush ran against Bill Clinton.