This page of the guide to Borderlands 3 has information on how to use emotes, animations useful for communication between the members of the team. The way in which the player can use emotes isn't very intuitive.
To use emotes, you need to hold the pause button - the Options button on your controller (default settings). The entire hud disappears and the game will show a drop-down menu with emoticons - see the picture above.
The drop-down menu allows you to use emotes. There are also separate options to point a target for your team and to challenge another player to a duel.
You can set emotes in a QuickChange vending machine - go to the Emote tab. You will see the list of all available emotes.
You can purchase new emotes from Crazy Earl in the Sanctuary - his shop is next the ship's hold. Open the shop's menu and select Emote category. Sadly, new emotes cost eridium (the premium currency), to credits.