Lutris
Lutris is a Linux gaming app that primarily provides installers for games not available on Steam. It can also act as a single app for your whole game library.
You can also install game launchers like the Origin Client, Battle.net, and the Epic Games Store. Lutris also lets you import game launchers from various sources, such as GOG, Humble Bundle, and Steam.
Installing Lutris
You can typically install Lutris through your distribution's app store (e.g. GNOME Software, Discover, Pop Shop).
For example, on Pop!_OS, go to the Pop Shop and search "Lutris".
Then, click install.
See Lutris's download page for more ways to install it.
Installing a game through Lutris
The most important feature of Lutris is that you can install games and launchers not found on Steam using install scripts. If you want to use an install script, select Lutris in Sources and click "Community Installers".
Then, in the top bar, click "Search games" and put in the name of the game you want to install.
Here, we'll click on the game with the photo, and click "install". There may be multiple install scripts options, so pick what seems most standard. If your first choice doesn't work, you could try the other ones.
You don't have to use the Lutris client to find install scripts - you can look up install scripts on Lutris's website.
Installation
Typically, when installing a game, Lutris will ask if you want to download certain files.
Review the files to make sure everything looks okay, and then just press continue. Don't worry about the individual download buttons.
Install games that you own on GOG, Humble Bundle, and Steam
Login to these stores by navigating to their tab in Sources, and logging in to them by clicking the Connect icon.
After logging in, Lutris should display the games you own from that store.
Troubleshooting
This video guide explains how to install a game with a Lutris install script in more depth.
If you encounter an issue installing your game, look up the game's name on the Linux Gaming subreddit to see if anyone's had the same issue and solved it. You should also check whether the game currently runs on Linux in the first place.
Runners
Sometimes, like if you download a game from GOG, you'll have a folder with all the game components. In this case, you'll use a runner to install it. A runner is a method to start your game. Lutris supports several runners: Linux, Steam, and Wine.
Linux is for native Linux games. Steam is for Steam games - though, using Steam itself to run those games is easier. Wine is what you want to use for every other game.