Installing asphalt roofing should be left to professionals, but repairing a couple of damaged asphalt shingles is easily done.
Individual shingles that you repair and re-install are typically 36 inches long and 12 inches wide with tabs cut to every 12 inches halfway through the width of each shingle.
The base of the tabs contains a line of soft tar that will soften in the heat and will help in sealing the shingles to the roof.
If you have damaged shingles, it is important that you do repairs and replacement as soon as possible as even a small leak can do considerable damage to the house and these repairs may extend the lifetime of a roof considerably.
There are plenty of videos that can explain the process of how to replace a damaged asphalt shingle. My favorite is from Ask the Builder available through youtube:
To do the replacement, you have to first remove the old asphalt shingle. To dot that you normally have to tap a pry bar under the damaged shingle just above the gap between tabs until you hit a nail.
You need to center the tool's nail-pulling slot on the nail, and press down on the other end to pry the nail partly out.
To get the nail rest of the way out, you need to place a block of wood under the bar just down the roof from the nail to act as a fulcrum.
You need to remove all nails that penetrate the top edge of the damaged shingle and usually also secure shingles directly above the damaged shingle.
When the damaged shingle is unsecured from nails, you need to pull the damaged shingle out and put the replacement shingle in the same place.
The alignment of the replacement shingle needs to conform with the other shingles. Then, secure the shingle in place and lift the shingle above it as needed to nail the new one into place with roofing nails.
You also need to replace the nails that you removed from the shingles above once you've secured the replacement shingle.
You also need to dab roofing cement over the nail heads that you could not remove without breaking the shingle.
Finally, spread roofing cement under each shingle tab that you lifted during the repair, and press these shingles down individually to adhere them and secure their placement to the one below.