On a board of a game with point-to-point movement, there are certain spots that can be occupied by markers or figurines, e. g. cities on a map. These points are connected by lines, and movement can only happen along these lines. It is not enough that two points are next to or close to each other; if there is no connecting line between them, a player cannot move his or her piece from one to the other.
Examples for point-to-point movement: Nine Men's Morris, Kensington, Friedrich.
Non-intuitive example for point-to-point movement: Risk. While Risk appears to be an Area Movement game like Axis & Allies, it is actually a point-to-point movement game, due to impassible water areas requiring overwater line connections (e.g. Japan, Brazil, Australia).
Pictured above: Point-to-point movement in Friedrich.