The Congressional District Method is a way that states can appropriate their electoral votes in a United States Presidential Election.
In the Electoral College, the states decide who gets to be president, based on the amount of representatives and senators they have. Since the early 19th century, it has been customary for states to decide this by the majority or plurality winner of that state's popular vote, (or at least the enfranchised portion). However, the federal government is (theoretically) agnostic about how the states make the decision. There is nothing to prevent a state from deciding, for example, that the Governor gets to give the electoral votes to whoever he prefers.
The only exception to the winner-take-all popular vote method of the states is the "Congressional District Method". This is only currently practiced in two states: Nebraska and Maine. In this method, the states' congressional districts are separated, and the presidential candidate who wins the popular vote in that district gets its electoral vote. The overall winner of the state gets two electoral votes.
Nebraska and Maine are both small states, and their electorates are fairly uniform across congressional districts. Maine has been using this method since 1972 and Nebraska since 1992, and only once, in 2008, has a state split its electoral votes, when a single district in Nebraska voted for Barack Obama.
Thus, the chance of either state splitting its electoral vote is fairly small, and the chance that it would make a difference in a presidential election is even smaller.
There is occasional talk of other states going to the congressional district method, and while it would have some advantages for some states, the chances are that it would be so politically disadvantageous for some groups and factions that it is not likely to happen. Also, the inertia of custom makes it unlikely that many (or any) other states are going to out of their way to change their methods. So for now, the congressional district method is more a curiosity of the American Electoral College system.