Fractions are smaller parts of whole numbers: one-quarter, one-tenth, one-half, and a faction is a smaller portion of a larger group that breaks away from it. A faction might take a fraction of the people from a large group and start a new group.
The Latin source of the word faction means to "do" or "make," though a faction can seem to be a taking away or un-doing. Politics is one area where faction gets a lot of use, because a political party often starts with a large group but has some members who disagree with a belief or direction. These members separate from the larger party and become a faction of their own, with many of the same beliefs as the original group but with a few new differences, too.