Well, here's how it works: When you are on a long adventure (remember you can only have up to 3 partners with you) you go to the pause menu and select Companion Mode, it'll give you a list of befriended Pokemon that you can choose to play as. When you choose one you are able to control that Pokemon back at your Paradise and walk around Paradise (but you can't go to the town, and as a result you'll notice all the shopkeepers in your Paradise while you are in Companion Mode) and even go on a quests or buy items.
If you are in Companion Mode you can also upgrade your Paradise.
A companion can take money or items from the Deposit Box to use, but when exiting Companion Mode the items and money will be returned to the Deposit Box (obviously).
It is a good idea to have a companion get a particular item you need so you may get it from a Deposit Box on your jouney.
A companion can also spin Victini's V-Wheel and can change the V-Wave to something that will help you out. You should also remember that going out on a quest takes one full day in the game, so if you change the V-Wave for the main player or someone on your journey and then quest in Companion Mode on the same day the V-Wave won't be there anymore after the quest.
The first chance you get, you should put treasure boxes in your Deposit Box when you're on an adventure, then have your companion open the box by going to the Box Buster (Ramphardos)
and then you could send the items back to the player if they are useful (on rare occasions the boxes contain badges).
So in Companion Mode, you can control one of your friends back at home, and they can do anything you can regularly do at Paradise, EXCEPT leave Paradise (unless going to a dungeon).
I think I covered it all, if there's anything I forgot just tell me and I'll edit it.
Hope I helped!
Source: I played Companion Mode many times while on my advnture to the Great Glacier