A* is a best-first search, meaning that it solves problems by searching amoung all possible paths to the solution(goal) for the one that incurs the smallest cost(least distance, shortest time, etc.), and among these paths it first considers the one that appears most quickly to the solution.
At each iteration of its main loop, A* need to determine which of its partial paths to expand one or more longer paths. A* selects the path that minimizes
f(n) = g(n) + h(n)
where n is the last node on the path, g(n) is the cost of the path from the start node to n, and h(n) is heuristic that estimates the cheapest cost of path from n to the goal. The heuristic is problem-specific.
a sample of A* search
A* is admissable and considers fewer nodes than any other admissable search algorithm with the same heuristic. This is because A* use an "optimistic" estimate of the cost of a path through every node that it considers -- optimistic in that the true cost of path through that node to goal will be at least as great as the estimate.
Dijkstra's algorithm is a special case of A* where h(x)=0 for all x.
BFS(Breadth-First Search) is a special case of Dijkstra's algorithm where all edge weights equal to 1.
What set A* apart from greedy best-first search algorithm is that it take the cost/distance already traveled, g(n), into account.
Reference:
A* search algorithm, wikipedia
Dijkstra's algorithm, wikipedia