You notice a piece of shingle in the yard. Looking up to the roof, it is apparent that a recent storm has damaged a few shingles in the same area. It is unlikely that you will be able to get any of the local handymen out for a job this small. Damaged shingles are not that hard to remove and replace.
Get shingles and roofing nails.
Take a piece of the shingle tab with you to the home center to matching shingles. Count the number of shingles that you will need. You will probably have to buy a 1/3 square bundle that will have more than enough shingles for a small patch. If your roof is more than a few years old, you will need to just get something close to the same color. You will need about 5 or 6 times as many nails as shingles. Get nails at least 1 1/4 inches long in case you have to go through an extra layer of shingles from an old roof.
Use a ladder that reaches a couple of feet higher than the edge of the roof.
If you are doing this job alone, tie the top of the ladder to your guttering or something nearby to keep it from sliding away. If your roof is more than 5 feet from the ground, it is a good plan not to have to jump off of the roof because your ladder did not stay put. Having a secured ladder is also safer for getting onto and off of at the roof line. By having it a little taller than the eaves, you can grip the top of the ladder to make your transition times simpler.
Carry the new shingles and nails up to the roof.
You should not attempt to climb the ladder if you are overloaded. Unless you are used to this type of work, it is best to break open the bundle and carry the shingles up in smaller quantities. You will need a good claw hammer and a flat prying tool. Having some type of wider putty knife is not a bad idea.
Move the shingles, nails, and tools to where they are needed.
It is easier on your roof if you can minimize the trips that you make back and forth across it. By getting everything nearby at the beginning of the job, it will make fewer trips. It is a good plan to put the tools and material above you on the roof if possible. This way, if they decide to try to slide away, you will be able to just catch them as they make it to you.
On steep roofs, you will need to secure yourself.
If the roof pitch is greater than 6/12, it is best to either attach a plank to brace against to keep from sliding or tie yourself to something on the roof. Should you start to slide on a steep roof, it can be difficult to stop before you reach the edge of the roof. Take whatever precautions are necessary to be safe.
Remove any pieces of the damaged shingles.
Use the putty knife to slide under the tabs on the shingles to break the bond with shingles below and above it. Use care not to damage these shingles. The shingle above the one being replaced will need to have the nails removed from it so it can be raised enough to put the new shingle into place. This will require that you loosen the tabs of the shingle that rest on it. The nails will be right above the slots on the upper shingle. Use the flat bar to get under the heads of these nails to lift them out.
Insert the new shingle into place.
Gently hold up on the upper shingle and slide the new shingle underneath it until the tabs of the upper shingle cover the flat area of the new shingle. Drive nails into the new shingle on each edge and two evenly spaced in the middle just above the tabs. Lower the upper shingle and repeat the process on it. If they are not damaged, use the same holes to insert the new nails. If the openings are damaged, drive the new nails just to the side of the old openings. Repeat this process until all of the replacement shingles are installed.
Remove the tools and shingles from the roof and make your exit.
On a single story house, it is a simple matter to just drop all the things that need to be removed from the roof over the edge. Make sure that there is no one to be harmed below before hurling things to the ground. You may want to consider carrying any new shingles down to keep them in good condition for potential use later. Untie the ladder as you exit the roof. Lower the ladder and clean up the the area, and you should be finished.