An algorithm a memorized sequence of moves that has a desired effect on the puzzle is called an "algorithm". For example, an algorithm is used to cycle 3 edges, or 3 corners. A commutator is also considered an algorithm. Specific notation is typically used to display algorithms which makes it easy to see and understand. In speedcubing, algorithms are typically memorized in order to solve specific positions in a short amount of time (recognition of what needs to be solved, recalling the algorithm needed to solve the position, and executing the algorithm to solve the position). The execution of an algorithm can be made more fluid with the help of finger tricks.
In mathematics and computer science, an algorithm is a step-by-step procedure for calculations. When ported over to speedcubing, it's very similar.