This is everyone's image of showing the dog an old piece of clothing and it takes off following exactly where the subject went. These dogs are said to scent discriminate, meaning that they differentiate between different people's scents and only follow the subject's scent (you hope!). Trailing dogs do exist and I've gotten to see a couple, but they aren't very common. I'm not sure why.
Wide Area Live Find
These are the dogs most commonly called out for recently lost hikers. They cover large areas roaming out from their handlers, though hopefully staying in sight, until they find someone and then bring their handler over to that person. Note that these dogs will find any live person in the search area be they your lost person or not.
Human Remains Detection
The sad reality is that not everyone who needs to be found is still alive. Sometimes there are suicides, sometimes a body is dumped, sometimes there's a house fire and authorities want to account for all missing persons. Sometimes you get to look for the murder weapon.
Human Remains Detection or HRD dogs can work a wide area like live find dogs, but are often worked in a much smaller area trying to find small bits of scent articles and then alerting such that people can carefully sift through those few square feet. It's a job that involves less hiking and people don't like to talk about that much, but it's common and helps bring closure.
There are certainly other types of search dogs. Rubble/Disaster dogs and shoreline dogs come to mind right away. Different areas use different terms and divide up the work differently as well.
So what am I training Beau in?
If my team did trailing I'd be tempted to start with that given that she loves following tracks. However, we don't do that so I'm starting her with wide area live find. It's a good fit for her energy level and seems to be really self-rewarding for her. Once we have that down and are certified, probably about two years out, then I'll start cross training her in HRD since that's a lot of the calls.