Faction Rallies are sporadic events on Destiny 2 where players can pledge to one of three factions and get tokens for them that can be exchanged for rewards such as gear and legendary shards. At the end of the event (duration seems variable but they tend to last for around a week) a winner is chosen. Players who pledged to the winning faction will be rewarded with their faction’s “Winner Offering” unique weapon, while the rest will be able to purchase said unique at the price of 50k Glimmer.
I’ve made Imgur albums for all faction weapons so far. Only 7 (out of 8) are available at a time when claiming a reward, and the list rotates. I’ve noted the possible Perks of each weapon on the description so you can get an idea of what the default Perk is biasing the stats, and what to expect of it overall. Hopefully that’ll help you choose a faction better instead of running back and forth between the guys. Feel free to point out if I’ve messed up somewhere.