The "Adjusted Gross Score" is one of the ways that the USGA uses to make sure a player's handicap represents his or her potential. Rather than use an unusually high score on a hole to include in a portion of the handicap calculation, for handicapping purposes, that high score is adjusted downwards according to the player's Course Handicap. The USGA Handicap Manuals says...
An "adjusted gross score" is a player's gross score adjusted under USGA Handicap System procedures for unfinished holes, conceded strokes, holes not played or not played under the Rules of Golf, or Equitable Stroke Control.
Reformatting for clarity, the above statement would look like this:
An "adjusted gross score" is a player's gross score adjusted under USGA Handicap System procedures for...
Much of the time what we're really concerned with is the adjustment to scores using Equitable Stroke Control. Equitable Stroke Control specifies that, for handicapping purposes, a player has a maximum number s/he can put down for any hole depending on his/her Course Handicap for the course/tee combination s/he happens to be playing.
The scorecard below shows both gross score and adjusted gross score for each hole. In this example, the scorecard is for a player with a Course Handicap of 4 on the course/tee he is playing. Equitable Stroke Control requires that he post a maximum of double bogey on any given hole, if his Course Handicap is 9 or less for 18 holes. On holes 11, 12 and 13, his gross scores were more than a double bogey, so - for handicapping purposes - his adjusted gross score is adjusted downward to 5, 7 and 6 respectively, or double bogeys. For competition purposes, each of those same three holes is counted as a triple bogey...ugh!
For each course/tee combination you will have a calculated Course Handicap. Knowing your Course Handicap, you can look up the Equitable Stroke Control table and determine the highest adjusted gross score you are allow