First of all, a sweeper is a Pokemon that goes in there and kills everything in sight. It might have moves such as Dragon Dance, Calm Mind, Agility. Sometimes, it may just be a powerful Pokemon with a Choice Scarf or a Life orb.. A sweepers job is to try to kill as many Pokemons as possible.
Im gonna take a definition of off Smogon, as it explains what it is very well...
A Pokemon who uses physically offensive moves to do damage and bring down an opponent's team. Many carry Attack boosting moves, like Swords Dance or Dragon Dance.
So, the idea here is that you send out a strong Pokemon with a high Attack stat. Many common Physical sweepers are: Gyarados, Terrakion, and Scizor.
Again, Smogon has a very good definition for this...
A Pokemon who uses specially offensive moves to do damage and bring down an opponent's team. Many carry Special Attack boosting moves, like Nasty Plot or Calm Mind.
Basically, a Pokemon with a very high Special Attack stat is most likely a " Special Sweeper". A common example would be: Latios/Latias, Thundurus-T, and Volcarona.
Ok, something you didn't mention, but falls in this category. This is known as the Mixed Sweeper
A sweeper who uses both physical and special offensive moves to do damage.
So, these Pokemons carry the job of 2 Pokemons in 1. As noted, they carry both physical moves and special moves. They usually rely on items such as Life Orb or Expert Belt. A Prime example of this would be: Infernape and Kyurem-B.
To learn more about other roles, visit this page to learn about the roles a Pokemon may carry.