There are several different disciplines that SAR dog teams may be trained for:
Trailing dogs are worked on-leash. Before starting, a scent article belonging to the subject is presented to the dog. The dog will then attempt to follow the residual scent trail left by that particular person as they walked from the place last seen. Note that the scent trail may not coincide exactly with the footsteps, so trailing is different than tracking that follows footstep to footstep.
Airscent dogs are worked off-leash. Usually they do not require a scent article; they are trained to range away from their handler and search for human scent in the air and then home in on the person generating the scent. When a person is located, the dog will communicate the news to their handler in some way and will make sure the handler finds them too.
Cadaver dogs are trained to locate the scent of human remains. When remains are detected, they will indicate in some way where the source of the scent is strongest. Cadaver dog teams may further train to locate bodies underwater, usually from a boat. This can be difficult because the dog doesn't have the mobility to home in on the source of scent as it can on land. So the handler needs to read their dog's behavior and direct the boat driver accordingly.
Avalanche dogs are trained to locate people that have been buried in snow, and will help dig them out.
Disaster dogs are trained to work on rubble piles such as from building collapses. The handler may not be able to go where their dog does, so there is a lot of emphasis placed on training agility and directional signals. FEMA task forces train these dogs teams to be deployed on disaster missions.