This is "Wonder Trade" from Pokémon X, Y, Sun and Moon but under a new name. Here, you can offer one of your Pokémon to trade and receive a mystery Pokémon from a completely random person.
How to Trade
Starting a Surprise Trade works similarly to starting a regular trade (or a battle). First, you need to open the Y-Comm by pressing the Y Button while on the field. Next, you should decide whether to communicate locally (the default setting) or via the Internet. To toggle between the two communication modes, press the + or - Button while in the Y-Comm menu.
When you're ready, press the "Surprise Trade" button near the top of the menu. You'll be asked to choose a Pokémon from your boxes to trade. After selecting a Pokémon, you must save your game. The game will then issue a stamp for your trade request. Afterwards, it will automatically look for other stamps requesting for a Surprise Trade, to try and match you up.
While the game's attempting to hook you up, "Searching..." will appear in the bottom-left corner of the screen, next to the Y-Comm prompt. Once you've been matched up, that text will be replaced with "Trade completed!" Now open up the Y-Comm again and you'll be shown the results of your Surprise Trade. Hopefully it's a good 'un and not a stinker!
You've probably heard amazing stories about Wonder/Surprise Trade, where people get sent shiny, flawless IV Pokémon. That can happen, but the chances are strikingly low (and if it does happen, it's probably hacked or a clone). Most of the time, you're going to get other people's junk. Because there's no benefit to giving strangers actually good Pokémon.
Still, even if it's junk, occasionally, it can be useful junk. Perhaps it's a Pokémon you don't have, which you can add to your Pokédex. Or maybe it's a leftover Pokémon from somebody's breeding programme, which you can incorporate into your own. Before trashing a Pokémon you receive, check if it has a useful Ability, move or even the Poké Ball it's been caught with!
On the flip side, we strongly encourage sending Pokémon that you don't need, but could be useful for other people. Please don't send yet another bog standard Rookidee or Skwovet! Instead, try to send Pokémon exclusive to your version, starter Pokémon that you've bred, etc. There's no benefit to yourself of course, but--like we try to do with our guides--it feels nice to help others, right?