The least expensive identification is the simple tag that hangs from a ring on the collar. There are many styles, materials, and colors available, and your local pet supply store probably has a wide variety that can be engraved with the pertinent information. Literally hundreds of sources can be found on the internet.
Though a little pricey, there are some really neat high-tech options available today, such as the slip-on tag engraved with a QR (Quick Response) code and web address that point to your dog's registered profile.
It can be scanned with the smart phone of the lucky dog's finder to reveal details you have selected, such as your phone numbers, emergency contacts, medical conditions, or medications. When the code on the tag is scanned, an email notification will be sent to you -- and if that phone has a GPS, you will receive a location map of where the scan occurred. You can easily change the details as often as needed without having to remove the tag. I found this particular kind of tag, a QR Code Scruff tag, along with many other kinds, at www.DogIDs.com. (This is a must-have for the techie geek Basenji!)
Permanent microchip ID is another good option, now that most vets have scanners that can read the chip. A tiny transponder encoded with an identification number is injected between the dog's shoulder blades with a hypodermic syringe. When the scanner passes over it, the ID number is displayed, which can then be traced through a national microchip registry to find the owner. The microchip cannot be lost, altered, or destroyed. The cost will vary, but figure somewhere between $25 and $50. Some areas hold clinics offering low cost or free microchipping. Again, you must remember to register and update your information.
Tattooing is another option, but not as "user friendly" as a tag you can just glance at. Some dogs don't like having their ears, thighs, or bellies looked at by strangers (hey, neither do I!), and hair often obscures the number, as well.