I have a broken finger with a torn tendon. What happens during the surgery?

They get fixed! Surgery for torn tendons and fractured bones repairs the damages. You will probably get immobilized and need physical therapy once cleared by your surgeon. Bones get fixed by either pins or plates and screws to hold the pieces together, allowing movement before it gets stiff. Stitches are used to repair cut tendons, which allow scar tissue to fully heal the tendon edges. Good luck.
Fracture and tendon. It depends on the nature of the fracture and how to maintain the alignment of the bones (tape, splint, cast, pins, plate, etc) and the extent of the tendon rupture/tear and ability to achieve a primary or staged repair or transfer.