Definition: Gross score is, of course, every stroke a golfer has taken during a round, added up to a total score. Adjusted gross score is a golfer's stroke total for a round after accounting for the maximum per-hole scores allowed by the USGA's Equitable Stroke Control (ESC) guidelines.
All rounds turned in for handicap purposes must be played using Equitable Stroke Control guidelines, which set a cap on the score a golfer can turn in on any given hole when posting his score for handicap purposes. For example, you might have scored a 10 (yikes!) on the fifth hole. But based on your course handicap, the ESC guidelines might require that you count that score as a 7 when posting for handicap purposes. Your score is still the total number of strokes you played; your adjusted gross score is that total minus any ESC deductions. (See the equitable stroke control definition linked above for further details.)
Note that only golfers carrying an official USGA handicap need worry about adjusted gross scores.
