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.
With point-to-point movement, you do not have a division of the board into areas which can be moved out of or into freely (like with Axis & Allies). Neither do you have a board completely covered in squares (like with Chess) or hexagons (like with Tide of Iron) that allow unrestricted or nearly unrestricted movement in any direction and to any square or hex.
Unlike these counterexamples, point-to-point movement arbitrarily restricts areas on the board that markers or figurines can occupy, and it also arbitrarily restricts the ways that these points may be reached. Oftentimes, this allows for interesting strategies.
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.