Try to open the cabinet door by pulling the handle. If the door will not open, Stick the blade of a putty knife between the handle-side of the door and the cabinet. Pull the handle as you pry the door with the putty knife.
Spray penetrating oil onto the hinge pin. Give the oil a minute of so to do its work. Try again to open the door. If the cabinet door still won't open, tap the pivoting portion of the hinge with a mallet. This step is sometimes needed to break apart rusted or corroded parts of the hinge. Apply more penetrating oil. After another minute, try again to open the door. If the cabinet door still won't open, you need to remove the door.
Unscrew the hinges where they are attached to the cabinet. Remove the door. Unscrew the hinges from the door. Remove the hinges. Hold the hinge, using two pairs of pliers, one on each side of the hinge pin. Try to flex the hinge. Inspect the hinge to determine if it is still in working order and that the hinge pin is not bent or deeply corroded. The penetrating oil will also lubricate the hinge. If the hinge is still difficult to flex, replace it with a new hinge.