Short answers, pulled from the story.
Peter Hart, Nils Nilsson, and Bertram Raphael of Stanford Research Institute first published A* in 1968. Nilsson originally proposed using Graph Traverser for path planning; Raphael suggested combining traveled and estimated remaining cost; Hart contributed the theoretical concepts of admissibility and consistency.
A* was created as part of the Shakey project, which aimed to build a mobile robot capable of planning its own actions. It was originally designed to find least-cost paths, but was later shown to find optimal paths for any problem satisfying the conditions of a cost algebra.
An admissible heuristic never overestimates the true cost to the goal, which guarantees A* returns an optimal path. A consistent (or monotone) heuristic additionally satisfies a stricter edge condition, ensuring A* processes every node at most once. Without consistency, a node can be expanded an exponential number of times in the worst case.
A* keeps every generated node in memory, so its space complexity is roughly equal to its time complexity in the worst case. On large graphs, memory is exhausted long before time becomes the bottleneck. This led to the development of memory-bounded variants such as Iterative deepening A (IDA) and SMA*.
Dijkstra's algorithm is a special case of A* in which the heuristic function h is set to zero for all nodes. A* achieves better performance than Dijkstra by using a heuristic to guide the search, expanding fewer nodes on average. With a consistent heuristic, A* is equivalent to running Dijkstra on a graph with reduced edge weights.
Epsilon-admissibility is a relaxed optimality guarantee used by approximate variants of A. A path found by an epsilon-admissible algorithm costs no more than (1 + epsilon) times the optimal solution. Weighted A, Dynamic Weighting, and the XUP/XDP parabola variants are among the algorithms offering this bound.