A heuristic is a system that will usually provide a shortcut to a correct answer. Opposed to algorithms, which will always reach a correct answer, but often take longer.
To illustrate, an algorithm for solving an anagram could test every possible combination of letters (it could also ignore letter combinations that never occur in English). This could take a long time. A heuristic user would would cut out letter combinations that almost never occur, and might look for certain combinations first. They might also take contextual factors into account.
Humans nearly always use heuristics. You will solve the anagram YBO without sorting through all six possibilities. You can probably use a simple heuristic on th2yEnrveig to cut through over 30 million possibilities in a matter of seconds. On the other hand, you might be slowed down by one of your heuristics with IPMHYLOD, as there is an unusual letter combination in the word.
Most math is done by algorithms; for example, adding the digits of a number to see if they are divisible by nine is an algorithm, as it will always work.
Heuristics are commonly known as rules of thumbs.