The bureau members are not human; they are what some humans have at times referred to as "angels", though they don't accept the term nor agree with everything it connotes. They are given powers by the Chairman, who sets limits on their powers such as not being able to track people surrounded by water. They live for a very long time but are not immortal.
The most important of their powers comes via their hats and books, which aren't really their own powers: whoever uses these powerful items gains the powers they afford. Their hats allow them to make use of teleportation doorways - normal doors in the real world that, if opened while bearing a hat, open to a completely different area, perhaps miles away. All doors in the normal world have these alternative connections; the bureau members possess and memorize a map of these routes. This allows them to move through the normal world at great speed. Every door may also be used to go directly to the Adjustment Bureau itself (whether this is a building somewhere on Earth, or elsewhere, or both, is unclear). The books allow the members to track people in real-time on a representation of the Chairman's Plan, see how people may be deviating from it, and the magnitude of potential unwanted side-effects that their deviations will cause unless adjusted. It gives the members all the information they need to make adjustments that keep people on track with the Plan. Because of their book, their knowledge of the Plan for the people they track, and the sheer fact that they track people their entire lives, bureau members have extensive and intimate knowledge about those they track, including information on how they think. Much of this is because the person has been adjusted to think that way. As such, bureau members seem able to read the minds of those they track, though they deny this to be true. It is a combination of extensive information and inference.
The only true power the members have is some subtle form of telekinesis which they do not use often. They can, with a gesture, cause some small force that could, for instance, cause a person at close range to slip. They seem to reserve this ability for when absolutely necessary.