With identifying bodies from dental records, it's unusual to have no idea of who someone is. Either there's strong circumstantial suspicion and the coroner requires confirmation or it's a "closed" case (such as a plane crash) where you know the IDs of eg six people but not which remains belong to which person. Open cases can remain unresolved for years (like the Kings Cross fire) and rarely, people use the opportunity of a mass event to "go missing" which can confuse things.
If the coroner requires confirmation, there are a few primary methods which they will accept, these are DNA, fingerprints, odontology and (in the UK) uniquely serial numbered medical implants. Odontology is quick, cheap and reliable (can have an ID within the hour for less than £100) compared to the other methods, it also doesn't require a reference DNA sample which families can sometimes find distressing.
The police may request dental records of people believed to be missing and then it's a case of elimination based on missing teeth, restorations etc. Occasionally it gets into very fine detail if people have a full, unrestored mouth.
If a body is found and there's no indication of who it might be, sometimes the police will put a notice in the dental press, especially if there's something unusual about the teeth or mouth or if it's obvious it was in an active treatment plan.
When you're called to see a body that has no indication of ID, sometimes you can narrow down the demographic for example, age, sometimes culture, likely socioeconomic status, recent treatment, whether teeth were broken pre/peri/post mortem and so on. This is sometimes useful for the police.
When it's a closed case, it's simply a matter of looking at the ante mortem records and comparing them to the post mortem data. There's software that helps with this but i've not used it as I haven't been involved in a large enough case. It's relatively straightforward for cases of 5/6 people.
With an open case (such as the Morecambe Bay cockle pickers and some large scale IDs like those of migrants who die on their way to the UK), odontology is just part of a team that includes anthropologists and other forensic experts who have amazing knowledge of tattoos and jewellery and clothing and so on. These cases are exceptionally difficult compared to say, the London tube bombings as no-one knows their relative is missing save they left the country that they were usually resident in. They may not be reported as missing for years. With a mass UK event, there are special call handlers who work closely with the police to ensure the correct information is captured as quickly as possible.