AUF is an abbrevation for Adjust Upper Face or Adjust U Face that stems mainly from the freedom of the U-layer in many common 3x3x3 methods. It is an adjustment turn of the upper face, either U, U2, or U' (or possibly no move), often used to prepare the cube for an algorithm or sequence. Sometimes it is performed before one of the last layer steps, such as PLL or COLL, to speed up recognition.
A cuber might know how to perform an OLL algorithm if the stickers are facing a certain way, and rotate the U layer to match the pattern they're used to.
For example, the moves during the final (OLL) step of Erik Akkersdijk's 7.08 world record were U' FRUR'U'RUR'U'F' U2. The boldface moves are AUF moves, to adjust the cube for an OLL algorithm and a PLL skip (respectively).
Note that Adjust U Layer (AUL) would be a more accurate name, but the abbreviation "AUF" has become standard. It is generally assumed that the rest of the layer turns with a face.
Similar concepts to AUF exist in several methods, but none of them have a comparably widespread name.