A political fight caused by a dispute between branches of a government
over what exactly their powers are. A constitutional crisis, if it's accompanied by other divisions -- such as ethnic
or religious ones -- can degenerate into a civil war
very quickly. Responsible officials go to great lengths to avoid causing them.
Constitutional crises are common in newly established (or newly reformed) political systems. In Russia, for example, in October 1993, then-president Boris Yeltsin didn't like what his country's parliament was doing, and issued an order to dissolve the assembly and have new elections. Parliament kept meeting anyway, declaring Yeltsin's dissolution order invalid and unconstitutional.
The situation was resolved by a siege, after Yeltsin ordered the military to surround the parliament building and not leave until the parliamentary leaders were arrested.
More mature governments aren't immune. There was a short-lived constitutional crisis in the United States in 1973, when then-president Richard Nixon temporarily refused to comply with an order from the Supreme Court to hand tapes of Oval Office conversations over to Congress.
Constitutional crises are, by definition, never resolved by formal legal means, because they're based on disagreements over what the law is, not just what it means. Usually, they arise in circumstances a country's constitution simply doesn't cover. The resolution usually comes about by force, as it did in Russia, or by a demonstration of overwhelming public support for one side or the other, as in the Watergate tapes affair.