A very common failure is the pull rope. If you don't know what
to do, it can be a pain.
If it is just the rope that's broken, first thing is to take off the top cover. Usually it only takes three 7/16 bolts to remove the cover and get at the pulley. On Tecumsehies, you can often just remove the cap around the pulley with three smaller screws. What you need to do is rewind the pulley against the spring (like the rope was being pulled out) and then hold it there. You can rewind it by hand, and either clamp or hold it so the place where the rope ties to the pulley aligns with the rope hole.
I use a short piece of bailing wire to help push the rope through the hole. Once you have it through, tie a knot and seal the knot with a flame if the rope is plastic.
While you have the top cover off, lube the ratchet assembly on top of the flywheel. Usually just some oil in the top hole will do it.
Now put the top cover back on and remount the pull handle on the cord. Make it so it just pulls with a little tension. I buy 1/4" cord in bulk at WalMart.
A little tension on the cord. Knot and flame seal the knot. Now get back out there and finish that lawn!