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Computer chess has exploded in the last twenty years, with a computer world championship being established and many chess computer designers becoming quite profitable from their endeavors. Some of the programs hide their source code, but a great many are open source, most notably Fruit, which served as the basis for a family of chess engines.

What are some examples of heuristics or algorithms used by chess computers, and which have been most successful or are most popular?

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Algorithms mainly used are based on searching the state space. The problem is that the possible positions a piece can make in the next move is very huge. But it is finite. It is basically a search problem. There is a valid position a piece can take. That is there is a known start state and a know end state. There is no probability involved like the problem of tossing a coin.
A typical algorithm is to use brute force and compute moves for a fixed number of moves. MiniMax can be used in such cases. Alpha-beta pruning is also a well know algorithm used. Mostly the search spaces are represented using a tree based data structure.
Computer program also heavily use databases containing moves, openings, end-game positions etc.

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    The question asks about heuristics. The whole point of heuristics is to avoid the expense of brute force calculation of all legal moves. So an answer involving a brute-force approach is, unfortunately, not on topic. – jaxter Sep 16 '16 at 17:40

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