If you have an older model garage door opener like I do the chances are that the drive chain is connected to a length of strong aircraft cable which runs through a pulley at the wall end of your garage door drive rail.
If this cable becomes worn and snaps then there is a good chance that the door will fail to open or close and could possibly even let the garage door fall down with no drive control. Fortunately the springs on the door header are there to prevent any serious injury or damage in the event a driver problem does occur.
No matter what goes wrong with a garage door opener you should almost always be able to unlock the drive and lift the door manually. If it will not open even with the cable drive disconnected then there may be something lodged between the rails and a drive wheel. If it seems that nothing should be stopping it from lifting them it may be wise to contact a professional and ask them for advice over the phone before paying for a service call. I fixed one where a loose screw had worked its way into the side wheel which locked this wheel making it extremely difficult to lift the door. SO do a careful job of looking at everything before proceeding as this may save you time and energy on this job.
The drive cable can be purchased at any hardware store and taking a piece of the old with you is a wise way to start this job. First make sure the garage door is unlocked from the drive chain by pulling the safety unlock cord. Then find the ends where the cable is attached to the drive chain. These are usually crimped on with a special tool but you can use a set of two cable clamps in place of these crimps. The drive chain loops through the motor end and the cable is at the garage door end with a pulley to guide it around the end of the rail.
Keeping all this hardware in place while you do the repair is simple if you use some wood holding clamps or vise grips locked onto the rail so the chain does not fall off the sides and hang to the floor. This will definitely be a dirty job since there is usually a lot of grease on the rails and chains.
You can hold the chain with channel locks while you use a cut off saw to nip off the old chain. If you have some really strong and sharp cutters you can cut the cable this way. The worst case situation is to use a hacksaw which takes forever and tends to fray the ends. Once i am finished clamping the cable ends I use electrical tape to wrap the ends to keep things neat and help prevent the cable from fraying apart.
Most doors are lightweight and the driver is balanced against the springs that are wound tightly to assist you and the drive motor when lifting the door. If your door is too heavy this can cause premature failure of the cable and even the drive motor as well.