In the United States government, has there been cases that electoral colleges don't vote for the candidate the majority of their state voted for?

In 1972, when Nixon won every state except Massachusetts (so that even Senator George McGovern's home state of South Dakota didn't vote for him) one of the Nixon electors in Virginia, Roger MacBride, voted for John Hospers of California and Theodora Nathan of Oregon, the candidates of the newly founded Libertarian Party.

I think Nathan may have been the first woman to win an electoral vote, but I'm not sure. When Vice-President Spiro Agnew officially announced the results of the vote of the electoral college, he called her "Theodore" Nathan.

Perhaps it should be noted that the term "electoral college" is not official. It appears nowhere in the Constitution. It was imported from Germany, where, under the First Reich, the king of Germany was nominally chosen by an electoral college in which membership was hereditary. The Germans used the Latin phrase "collegium electorum".

The general idea behind the Electoral College is that the states pick Electors, they get 1 each for each congressmen and senator they have (iow: at least 3 for every state, more for bigger ones), and the Electoral College decides, based on majoriy vote, who is President.

Nits:

• It is up to the states how they pick their Electors. It would be perfectly OK for the Governor or the state legislature's ruling party to just assign them, if the people of that state would put up with such a system. It's been done in the past. Right now all states allow their citizens to directly vote for electors, and in all but two states the majority candidate gets all the electors. However, that's for the states to decide. Originally, almost no states directly elected their EC representatives, so it would be fair to say that in most early elections, the EC voted for candidates that had not been voted on by "the people".
• Electors are free to vote how they please. Half of the states try to legally require their electors to vote for who they promised to vote for, but Federal law has no such requirement.
• They might not get to decide. If there's no majority winner, the election goes to the newly-elected House of Representatives (but on a one-vote-per-state basis). This has not happened in nearly 200 years, and really could only happen in today's universe in the case of a 270-270 tie. (There's an even more breathtakingly unlikely scenario where the Senate could end up indirectly picking the President, but let's not get into that)

There has never been a case where the Electoral College AS A WHOLE, went against the electoral votes generated by popular vote.

That said, there have been cases where individual ELECTORS went against their state's vote. One example I can think of is in 1988, when the popular vote of the state of West Virginia went for Dukakis. One lady elector from that state cast her vote for Lloyd Bentsen (Dukakis' running mate), for President and Dukakis for Vice-President. So Dukakis actually got only 110 electoral votes instead of the 111 he was entitled to.

States Electors aren't really bound to the popular vote, there have been cases where the popular vote did not decide the winner-take-all voting system we have; I note this because Main and Nebraska do have a different system as Keith notes. That system divides up the votes, and while this typically matches the Popular vote it has not always been so.

I'm not clear if you are asking about specific states, or in general for the elections where the national popular vote did not match the Electoral College - which gave the election to someone else. There are not really that many cases of this:

• 1800 election of Aaron Burr and Thomas Jefferson ended up as a tie with 73 electoral votes each, which put the decision to the House of Representatives and the result of this was the 12th Amendment
• 1824 had a four way contest with Andrew Jackson, John Quincy Adams, William Crawford, and Henry Clay all gaining votes but not a majority, which put the decision of the election to the House of Representatives (according to the 12th Amendment). In this case John Quincy Adams received the majority of the House votes, even with Jackson getting the most electoral votes, in this case the election went where neither the popular count or electoral count showed a victor
• 1888 election of Benjamin Harrison was a case where Harrison had some small majorities in some of the bigger states, which gave him a larger Electoral College count which gave him the election
• 2000 election of George W Bush and Al Gore often notes that Al Gore won the majority of the popular vote, but the decision by the Supreme Court gave the election to Bush. There is plenty of evidence on who won the votes on either side, though I tend to side with the evidence that it was Gore, this was the year of the Butterfly Ballot and hanging chad's

Remember though, that early elections were often handled by Congress who promoted candidates, voting was at first limited to only landed males and extended later so comparing the votes between early elections and present day needs to take this all into account.

Sources:

US Election Atlas

Obligatory Wikipedia entry

University of Missouri-Kansas City Law Site

Yes

In 158 instances, electors have cast their votes for President or Vice President in a manner different from that prescribed by the legislature of the state they represented. faithless elector

Hat tip to @Keith Thompson, but I wanted to provide a more specific answer.

Most states choose Presidential electors based on the candidate who got the most votes in the November election, but not all do. In particular, Nebraska (which has 3 congressional districts and therefore 5 electoral votes) allocates 2 electors to the state-wide winner, and each of the other 3 to the winner in each congressional district. In the 2008 election, Nebraska cast 4 votes for McCain and 1 for Obama, who won a majority of the vote in one district.

Maine has a similar rule, but I don't believe it's split its votes since it established that rule.

But electors are not Constitutionally required to vote as they committed (though there may be penalties under state law). The Constitution leaves it to each state legislature to decide how electors are chosen. All states currently do so by popular vote, but that hasn't always been the case.

There have been a few cases of "faithless electors". Most recently, a 2000 Washington, D.C. elector refused to cast a vote to protest D.C.'s lack of representation in Congress, and a 2004 Minnesota elector wrote the wrong name on the ballot, apparently accidentally.