In AH, Fighter/Attack/Bomber are basically just scoring categories. Some planes can be flown only as fighter, some as fighter or attacker and so on (you can select scoring mode in hangar)
Scoring modes in a nutshell:
Fighter: Only hit's and kills on enemy planes are used for scoring purposes.
Attack: Hits on and kills of planes & vehicles, as well as damage on ground structures do count.
Bomber: Only damage to ground structures is used for scoring.
For more info about scoring, read
http://www.hitechcreations.com/Help-Section/Aces-High-Features/aces-high-help-scoring.html and take a look at the Scores & Ranks part of this website.