Heuristic Function is a function that estimates the cost of getting from one place to another (from the current state to the goal state.) Also called as simply a heuristic.
Used in a decision process to try to make the best choice of a list of possibilities (to choose the move more likely to lead to the goal state.) Best move is the one with the least cost.
It can also be defined thus as a function that ranks alternatives in search algorithms at each branching step based on available information to decide which branch to follow.
Example for Blocks World Problem:
Consider below mentioned Start & Goal states for Blocks World
h(s) = Number of places with incorrect block immediately on top of it