A stray dog is one that has, for whatever reason, become separated from his or her owner. If you are faced with the prospect of trying to catch a stray dog there are several factors to consider, the first being risk to yourself.
Stray dogs could be scared, a scared dog will react by biting. If you want to catch a potentially frightened dog, you must be familiar with how to read their body language and how to react to it correctly so you do not provoke a problem. Never have a child help you catch a stray dog, children often miscommunicate and send the wrong signals to a dog and will then get bit.
Another risk is that the dog is diseased, if it is behaving oddly do not attempt to catch it, this could be a sign of a dog with rabies or who is mentally unstable. Call a professional instead.
Dogs pretty much always respond to food. You can befriend the dog by offering it food, when it is close enough you can take a hold of its collar (if it is wearing one) or slip something around its neck for control. This is easily done if you have a leash, just run the hook end of the leash through the handle end and you form a loop at the handle end to go around the dogs neck, and can hold the hook end.
If the dog is shy but not threatening to bite, this may take patience, throwing small bits of food out to the dog, gradually bringing it closer. Always talk softly to the dog. If somebody else is approaching and they threaten to hamper your attempts in the same voice speak to them to back off.
Most dogs respond to being called "Here boy", show a friendly gesture by bending forward and slapping your upper legs with your hands, have an enthusiastic sound in your voice. Pretend you are calling your own dog. Try saying "Come".
You can lure a dog into your fenced yard then shut the gate behind it. You can do the same thing to get it into your garage. Put some food in the garage and wait. Canned food will have the best smell for attracting the dog. If the dog is an agressive one, and you have a situation where you can do this, tie a rope to the handle of the garage door, when the dog enters, you, from the safety of your home, can pull the door shut.
Most veterinarians, or animal shelters, have proper traps a person can rent. You can use these to trap smaller dogs, as they are usually used for cats, skunks, and raccoons. Similarly you can call a professional out to catch the dog. Dog catchers have special poles, often called rabies poles, that they can use to reach out and nap an aggressive, or unfriendly dog, with.
Most areas have somebody in charge of catching stray dogs, they are unsafe roaming around. If you are unsure who to call, a veterinarian can give you the correct phone number in this event.
Now you have it, what do you do with it?
Once you have caught the stray dog.
This dog is not yours to keep. It is a stray. Your next move is to find its rightful owner. Call your local shelter or pound (again if you do not know who to call, as a veterinarian). Legally you must report finding the dog, or must bring it into the shelter. If it has a collar, look for a tag that might identify its owner, if not, the dog might have a microchip which a vet could check for at no cost to you.
If the dog is in need of medical treatment, call the shelter and see which vet the suggest you take it to, since a stray dog is their financial responsibility, they should determine what vet it sees.
If you are intending on keeping the dog, ask the shelter how long you have to wait until they consider it legally yours. In most areas this is 14 days after you first report finding the dog.