Factions are groups of players within MCM grouped together under a common banner or cause.
Making A Faction
Factions within MCM are player created and governed affairs. The current site system is under renovations for supporting it within the MCM database. Therefore, making a faciton is as simple as making a group within second life. Figuring out what colors you want if any, and recruiting players.
From there you may wish to decide on any clan colors or asthetics if you want them (and make sure it doesn't conflict with any other known teams). Once you have that settled it's just a matter of getting on the field, winning fights, and making a big enough name of yourself that other players recognize you. Also you may wish to make use of the faction tempate page.