Good heuristics must be fast to compute, because if it takes so long to compute the value of a heuristic at a single node, it may have been preferable to have just expanded more nodes using a cheaper heuristic. For example, if the heuristic function is a breadth-first search to find a solution and its cost, then this is clearly too expensive to be useful.
Can often devise good heuristics by computing the cost of an exact solution to a simplified version of the problem. For example, in the 8-puzzle, if we relax the assumption about how a tile can be moved so that any tile can be moved in a single step from any position to any other position, then this means that a solution costs only the number of misplaced tiles since each misplaced tile can now be moved in one step to its goal position. This heuristic is admissible because at each move just one tile moves one position, so this is the minimum number of steps to get each of the misplaced tiles to their goal position.
Similarly, if we assume that tiles in the 8-puzzle are restricted to moving one square horizontally or vertically at a time, but we relax the assumption that only one tile can occur at a board position at a time, then each tile can be moved independently to its goal position, taking a number of steps equal to the Manhattan distance from its start position to its goal position. This leads to a heuristic which is the sum of the distances of the misplaced tiles to their goal positions. This heuristic is admissible.