A long time ago, the only way to provide identification for your pet was to purchase a name tag to put on a collar or take a sharpie and write the pets info on the collar itself.
Now we have microchips. A microchip is a very very small device that has a series of numbers and letters encoded into the chip. A special scanning device is needed to read the series of numbers and letters. When you pet gets microchipped, this chip is inserted under the skin with a needle and syringe supplied by the microchip company. The universal location of the chip is between the shoulder blades. Your pet will have a microchip "number' that will need to be registered with the company. There usually is an annual fee to keep your pet registered. Without this registration, the chip is useless. The microchip itself can cost $30 to $45.
The company I use for microchips is Home Again. There are other companies and it is up to the veterinarian or shelter to decide which chip to use. Home Again has other offers that can can be linked to the chip but they may cost extra. You can call Home Again at 1-888-466-3242 or visit their site at HomeAgain.com.
Another form of identification for your pet can be a tattoo. I do have a tattoo machine that I use frequently on feral cat populations that are getting spayed or neutered. A tattoo can be put inside the ear flap or on the belly or inner thigh of an animal. Most tattoos chosen by owners are their phone number. Tattoos have to be done under anesthesia and are therefore expensive unless it is done during some other anesthetic procedure, like a spay or neuter.