Repairing asphalt shingles can be done by yourself with little work, or by roofing repair professionals.
The repair work needs shingles that correspond to the existing shingles, being the same type, same color, and same size.
Lift the tab above the shingle that has been damaged and pry out the nails that hold the one you want to replace.
When on the roof, you need to remove the damaged shingle without damaging additional shingles with your repair work.
To remove the damaged shingle, you need to lift the shingle above the damaged one and pry out the nails keeping the damaged one in place. You need to do this without damaging the shingles on top or around the damaged shingle.
Then, snip off the top corners of the replacement and slide the replacement up under the existing shingles, being careful not to tear the roofing felt.
Finally, you need to nail on the new replacement shingle, placing the nails under the shingle that is above. Also, when the shingle is in place and nailed, you need to daub the nail heads with roofing cement.
There are several videos online that show you how to do this, so that you don't need to rely on reading about the replacement process alone.
My personal favorite video is from the website Ask the Builder: