A party in FeralHeart is pretty much like a private chat/closed conversation between chosen participants.
See it as a whisper function but for several users instead. If you are in a party with a group of users, ONLY the users within the party are able to read what the members inside the party say in the party chat that's active. Party chats are really useful for private conversations between you and other user(s).
You can easily start a party with other players by double clicking their character and then select "party" on the window that appears, that will send them an invite to a new or already existing "party" chat. If you receive an invite to a party the two chat bubbles icon in the bottom right corner will have the number "1" over it and become highlighted.