A "Player Group" is an entirely out-of-game organization - a group of players that play together and organize themselves using third-party tools (eg. Discord). To "join one", find their website, Discord channel for forum recruitment thread and comply with whatever arcane rules and requirements the group might require of you.
A "Player minor faction" is a faction which a Player Group requested to be added to the game. A Player Group needs ten members before it can qualify to submit a request for a PMF. Although a Player Group requested it to come into existence, there is no way for either the Group or individual players to directly affiliate with or "join" a PMF. It's just a matter of supporting that PMF though your actions (taking missions for them, blowing up Wanted ships in their jurisdiction and handing in the bounty vouchers to them, selling exploration data at starports they control, that sort of thing).
A "Squadron" is a clan-like in-game organization of players. A squadron might, or might not, be affiliated with a minor faction (either a PMF or a regular NPC minor faction). A squadron might, or might not, be linked to a Player Group. You join them via the Squadrons tab on your status screen.
A "Private Group" is a game mode in between Solo and Open - anyone can start their own Private Group, and you Join one already started by clicking on the Private Group option when you start the game, and filing a request to join with the group's creator. Some very large Private Groups with thousands of members exist, mostly to operate as a non-violent PwP environment; such a Private Group has rules where you promise not to shoot at or otherwise grief other group members while in Private Group mode, unless in specific circumstances (such as Combat Zones). Mobius and Fleetcomm are two large PwP Private Groups that I am aware of.