Their name is a reference to the Sixth Pilgrimage, of which all current Sixers are from. However, not all humans who came through the portal on the Sixth Pilgrimage are Sixers. Some of them stayed at the colony when the Sixers first went OTG (Outside the Gates).
As revealed in the episode "What Remains", the Sixers have a way to communicate with someone in the year 2149. They also have a medicine to treat Sincyllic Fever, with which they had been treating Skye's mother until she was rescued from the Sixers' camp by Curran and taken to Terra Nova.
The Sixers were sent to Terra Nova via a rigged lottery to try to overthrow Commander Taylor. They, along with Lucas Taylor, are trying to create a portal to be able to go back to the future so that they can use Terra Nova's resources. If Mira succeeds, she can see her daughter, Sienna, who is still in 2149.