In the context of "Harrison Bergeron," what makes a person normal or abnormal?
Kurt Vonnegut's short story "Harrison Bergeron" is a commentary on the future of mankind in a world where everyone is equal. There are amendments to the Constitution which now require everyone to be equal--another way of saying "normal." Everyone is to be the same, and to ensure it, various handicaps are given to each citizen to offset any talent or ability. This is an attempt to make everyone normal; instead, the result is a world of mutants who wear weights and rubber noses and thick glasses--all designed to take away the talents and gifts with which each person was born. This is, supposedly, a world in which everyone is normal.
In contrast, the "abnormal" character is Harrison Bergeron. He is clearly an especially gifted young man who is now, by law, encumbered with outrageous handicaps. He appreciates beauty, he refuses to be bound by his handicaps, he wants to dance, and he wants to soar. His tactics in trying to make that happen are rather outrageous, but he is bold and brave in wanting to break free of what is binding him.
In this futuristic world, then, "normal" is being weak and trapped and handicapped; "abnormal" is being willing to fight the shackles and aiming for something beyond "normal."