Roofs of garages, like any type of roof, can break down over time and will need a specific type of covering to keep out rain or other elements. So laying a covering over damaged places of the garage's roof should be done to protect the inside of the garage from the elements while it is being repaired.
A properly secured tarp over the part of the garage roof in desperate need of replacement is a good temporary solution to protect the roof while you are repairing it. As tarps can be easily blown off the roof, it will need to be secured to the roof in some way which will hold it in place and keep any excess water out of the hole. If you don't mind putting holes in the tarp, heavy duty industrial staples can be used to hold the tarp in place against the roof shingles. If putting holes in the tarp doesn't seem like a good idea, try weighting down the edges to secure the tarp over the hole in the roof, taut against the surface.
There are specific things which should be used to properly cover the roof for an extended period of time. Over the panels of wood which have been used to cover the rafters of the garage's roof itself, roll out roofing felt, which can be stapled or nailed into the roof boards. Along the edges, aluminium flashing strips should be attached with nails or screws. This will take most of a day to do, depending on the state of the garage's roof. If you need to take a break or stop for the day before you can attach the shingles over the roofing felt, be sure to cover it entirely with a tarp or waterproof piece of plastic.
Once the roofing felt and aluminium flashing has been affixed to the roof, attach shingles with roofing nails. Try to do this on a warm to hot day as the tar holding the shingles together will expand and melt slightly, making the shingles attach themselves to the roof more completely. Make sure the shingles are all laid out in a straight line, so that they will shed water properly and look aesthetically pleasing to the naked eye. Of course, you'll really know how well you did this the first time it rains and the roof does or does not leak.