Finding a valid path is a search problem
At its core, motion planning starts with a start configuration, a goal configuration, and a set of obstacles or constraints, and asks the planner to find a valid path between them, one that avoids collisions and respects the robot's physical limits like joint ranges. This is fundamentally a search problem: somewhere in the space of all possible paths, there exists at least one that works, and the planner's job is to find it without checking every possibility, because there are far too many to check exhaustively.
Often 'valid' isn't good enough on its own, and the planner is also asked to find a path that's best by some criterion, shortest distance, least energy, or smoothest motion. That turns the problem into optimization on top of search: not just any path through the maze, but the shortest or most efficient one. These two goals can pull in different directions, and real planners have to balance thoroughness against how long they're willing to search.