The largest faction being Fatah?
I know from the fact they sometimes kill each other that Fatah and Hamas don't see eye to eye, but to be honest in the Western media Palestinians are usually portrayed as a single homogeneous group - they're either all unrepentant terrorists or all innocent victims of Israeli oppression, depending on what you're reading. That makes it difficult to identify which factions are more peaceful and progressive and which are more violent.
I guess I should give a general picture of some major points here.
The PLO has been the dominant force in Palestinian politics, it also encompasses several groups inside it or factions, the biggest one being Fatah.
Hamas has not been affiliated with that movement, and only in recent years has it emerged as a counter Palestinian faction to Fatah.
today the Palestinian territories are divided. before that the PLO and Fatah ran all of these territories, meaning both in the Gaza strip and the west bank. after the rise to power of Hamas in 2006, the result was armed clashes between Hamas and Fatah, and that left Hamas running the Gaza strip and the Fatah running the west bank.
in official negotiations, which have included the Americans in great deal, Israel and the PLO have arrived to mutual recognition, and in 1994 after the Oslo Accords these negotiations paved the way for the creation of the Palestinian Authority. a self governing body over the Palestinian urban centres. Fatah has been considered a legitimate body at large in this regard, as engaged in negotiations with Israel. while Hamas was and still is considered an Islamist organization, which is still branded as a terrorist movement by various countries around the world including the EU.
of course this only simplifying things. as there are more factions within the PLO itself, some can be said to be more moderate while others can be said to be more extremist. in addition there are other radical movements in the Palestinian territories with their own interests.
the different Palestinian factions have been involved in conflicts in Jordan, Israel, Lebanon and fighting also among themselves in inter-Palestinian clashes.