Is it possible for the party of the president and the majority party of the congress to be different parties?

Yes, but the majority party (not of congress- just what is considered the majority party according to laws) is determined by the executive branch aka the president.

The most recent example, while Bush was still in office the democrats took control of Congress, but the majority party of the country was still considered Republican. Currently our President (Obama) as well as the majority of Congress are Democrat but if at their next election Congress shifts to a majority of Republicans the country's majority party would still be considered the Democrats, because of the president

Added: The initial answer is incorrect. The majority party in Congress is ALWAYS the numerically superior party, regardless of WHO the President is. The President has nothing whatsoever to do with naming the majority party, the voters decide who it will be by putting more members of one party in office, rather than the other.