In Kurt Vonnegut's "Harrison Bergeron," equality has been achieved by reducing each citizen to the social lowest common denominator. Specifically, this has been done through the passage of the 211th, 212th, and 213th amendments to the U.S. Constitution. In practice, this means that each individual who possesses some trait that makes him or her better than others—intelligence, looks, athletic ability—must wear a handicap that brings him or her down to a lowest common ability.
An example of the above is the relationship between George and Hazel Bergeron. George's intelligence is "way above normal," so he has "a little mental handicap radio in his ear" that "would send out sharp noise to keep people like George from taking unfair advantage of their brains." Hazel, however, has "perfectly average intelligence, which meant she couldn't think about anything except in short bursts." As a result, she wears no handicaps.
In the opening paragraphs of the story, George and Hazel are watching television and Hazel is crying, but, because of her natural short-term memory, "she'd forgotten for the moment what [the tears] were about." George also forgets what he is watching, but that's because of a buzzer that rings in his mental handicaps, which caused his thought to flee "in panic, like bandits from a burglar alarm."
While we might see equality as an idea in which people are allowed to raise themselves to the same social standing as others, Vonnegut uses the term ironically as a warning against allowing ourselves to be reduced to an equal unthinking place.