To answer this question I'd start with a distinction of a project management as a role, one of crucial ones even in one-man projects, and project manager as a title.
Now, you will basically always have a person, or a group of them, who are fulfilling project manager's role, however it is perfectly OK if neither of them has "project manager" in their signature.
Actually, contrary to others, I don't consider having a PM is crucial in agile projects. If we take Scrum, which is the most popular agile method we won't find project manager mentioned anywhere. It is because project management role in Scrum is spread over the whole team, with some kind of special position of Product Owner.
It means that Scrum teams, especially more experienced ones, can perfectly deal without someone who is called project manager.
Note: I use Scrum just as one of examples here -- I know Kanban teams which follows the same pattern as well.
Usually the reason to introduce project manager to agile teams is the way the customer wants to run a project. If the agile team is working with the client who prefers to have project managed the classic or formal or whatever you call it way you likely want to have project manager on your side -- someone who would be translating the way the team works to the way the client wants to see it.
If the agile team can work directly with the client using methods they want, e.g. product owner on client's side, demo after each iteration, client's active participation in prioritizing work for every iteration, etc., you probably won't find project manager in such team.
Another reason to introduce project manager to agile teams is when your organization wants to have projects run classic way for whatever reasons. Then again, you may want to have PM on the top of the team so you have someone connecting both worlds.
This is the most important job of PM in agile teams -- being a bridge builder, as some teams need those bridges to translate their own methods of work to something which is expected outside of the team. This is natural since agile methods tell us little about formal side of project management so they're falling short whenever it is needed.
UPDATE: Chris added in comment one thing which is worth stressing and hasn't been stated directly: PM in agile teams is less of command and control type of person as power to make many project-related decisions is delegated to the team. I wouldn't go that far to say that Scrum Master is agile version of PM, because is isn't, but still if you consider the team has more power someone has to have it less, and it's usually a PM.