oh hey, its a genre i play a lot
time for
JAVELIN'S LIST OF GOOD FPS GAMES AND WHY
Sorted into singleplayer/multiplayer for your convenience.
SINGLEPLAYER
Bioshock - You're stuck in an underwater retro-futuristic capitalist city and have to get out. Gunplay is meh, story and atmosphere are superb. You also get magic powers like lightning bolts and fireballs that you can throw at enemies. Make sure to mod it to remove mouse acceleration, otherwise aiming is terrible (designed for consoles).
Bioshock Infinite - You're stuck in a flying old timey racist city and have to get out. Gunplay is pretty strong, story and atmosphere are a bit confusing and divisive. Overall, I prefer this one to Bioshock because of more satisfying weapons, more interesting enemies, and more enjoyable characters, but they're both worth playing.
DOOM (2016) - Have you ever wondered what FPS games would be like if there was no reloading, you moved really fast, and could carry a million guns, like the originals? Doom does this and it's fun. Demons are invading and you get to murder every single one - with your own hands, if you'd like. Story is almost nonexistent (unless you spend time reading the logs, nerd) but the weapons are really fun and enemy variety is impressive. Gets a bit repetitive by the end, in my opinion, but still an enjoyable ride if you just want senseless violence.
Portal and Portal 2 - Imagine an FPS game, but you don't shoot anyone and just solve puzzles with a gun that shoots portals. Also there's a homicidal AI who constantly snarks at you in the background. Pretty fun games with a surprisingly entertaining story, and they're fairly cheap (and short).
Star Wars: Republic Commando - Command a squad of clone troopers and fight battle droids. Often regarded as one of the best Star Wars games, it's still pretty enjoyable today. One of the only games where your AI teammates aren't complete garbage and can actually do things.
Wolfenstein: The New Order - The Nazis won WWII. Time to fight back and murder them. This is quite possibly my favorite single player FPS of all time - great guns (with dual wielding), optional stealth, fun levels, a surprisingly good story, and of course there are plenty of Nazis to murder. The sequel wasn't quite as good though.
MULTIPLAYER
Team Fortress 2 - Class based gameplay at its finest. TF2 is a classic, it's been around for what feels like forever and is still enjoyable and humorous. Best of all, it's FREE and should run on almost any current computer due to its age. If you like multiplayer and you like FPS, you should definitely try it out.
Counter-Strike: Global Offensive - Modern tactical team-based FPS. Frustratingly hard, but still enjoyable. I played enough to know it's not for me, but if you're hypercompetitive you might enjoy it.
Red Orchestra 2 - WWII (Eastern Front) FPS, where big 30 v 30 teams compete on Attack/Defense. The devs put every effort into making it feel as realistic as possible, so there's no UI, no ammo counter (if you reload with a half-empty clip, that half-empty clip remains in your inventory if you try to reload later), you die in only a few hits (generally to a machinegun across the map you only saw a muzzle flash of). It's brutal, and you'll die over and over and over and over and it starts to earn a reputation as "PTSD Simulator". Definitely not for everyone, but once you get the hang of things it can be rather enjoyable - you can kill just as easily as be killed, after all.
Overwatch - Also known as Team Fortress 3. More classes, and lots of different ways to play - I appreciate that they made an FPS game you don't have to actually be able to aim to play. Has a few minor issues people complain about (class balance, not being able to pick a particular game mode) but is overall a well made game worth playing.
NOT QUITE FPS GAMES BUT I STILL WANT TO INCLUDE THEM
Fallout 3, Fallout New Vegas, Fallout 4 - FPS/RPG hybrids. Think Skyrim with guns in a post apocalyptic wasteland. Pretty enjoyable, New Vegas has the best RPG elements while 4 has the best FPS elements.
Deus Ex: Human Revolution, Deus Ex: Mankind Divided - FPS/Stealth hybrids. Sneak around in a cyberpunk future as a cyborg secret agent. Human Revolution's combat isn't all that great, forcing you to run stealth, but Mankind Divided can be played as an FPS, if you'd like (though you'd definitely be missing out on a lot of content).