A path puzzle is a type of grid-logic puzzle
in which you are asked to draw one or more paths on the grid according to certain rule
Typical rules include the number of squares from each row/column used, like in battleships (or a requirement that all cells will be used), lengths of paths, or specific endpoints of paths that must be joined to their matching endpoints.
Honey Way from the now-defunct Mind Aerobics was a good (if somewhat more complicated than usual) path puzzle.
Path puzzles made up a tremendous part of the World Puzzle Championship in 1999.