"Handicaps" are used in "Harrison Bergeron" to keep one person from having any advantage over another person. The Handicapper General is a person in this society who determines in which ways a person might excel over the general population, and this person is required to carry around physical "handicaps" in an attempt to make everyone "equal."
For example, people who are intelligent and freethinking are required to wear a device that emits loud and painful noises each time their thoughts exhibit free thinking. These noises are so intense that they immediately cause the person to abandon any free thinking. Both Harrison and his father wear this type of handicap.
Other characters are more beautiful than the general population, so they are required to wear hideous masks to cover up their beauty. People who exhibit great strength are required to carry around extra weights so that they cannot move as quickly or nimbly as other people. Consider the way this exhausts George, which is evident when Hazel comments,
"All of a sudden you look so tired," said Hazel. "Why don't you stretch out on the sofa, so's you can rest your handicap bag on the pillows, honeybunch." She was referring to the forty-seven pounds of birdshot in a canvas bag, which was padlocked around George's neck.
Harrison himself is even stronger and is forced to carry around three hundred pounds of scrap metal.
The contrasts in "advantage" are evident by how many handicaps people are required to carry. While Hazel isn't required to carry any handicaps and is therefore not naturally exceptional in any way, Harrison is forced to endure many handicaps in an attempt to "equalize" his physical and mental gifts.