The main surface and seating areas are probably leather. Most furniture manufacturers use a combination of leather and vinyl; leather on the contact areas and vinyl for the trim. This cuts manufacturing costs but still allows them to advertise the furniture as "leather". This kind of damage is common and unfortunately extremely expensive to reupholster. In this case, the backing fabric is still intact. Your best option would be to use a strong tape, preferably in a brown color, to give it a bit of support so that the fabric does not begin to separate.

Remove the back, the swivel platform, if equipped, and the rocker platform. (Other sites tell you how to do that.) Open the chair to the reclining position. With the chair upside down, pull the staples holding the upholstery and "undress" the area where you see all of the play while moving the arm in and out. This will be on the front vertical corner edge of the chair. Look carefully at the gap that opens and closes as you wiggle the arm. You will see the gap and staple legs that no longer hold the two pieces together. You will also see the remains of green glue that once held, along with the staples, the arm section to an inside panel. Carefully increase the gap so you can cut the staples with a pair of dikes. Since it was once glued, I used Gorilla glue in the gap and clamps to pull the gap closed followed by a 6 screw angle bracket on the outside and screws from the inside of the panel into the end piece of the arm section that was pulled away. Always drill pilot holes for the screws. Re-dress the upholstery and re-assemble the chair. Wish I could show pictures on this site so you could see exactly what I am talking about. It is now stronger than it was and you would never know I had worked on it.