Here are some popular choices:

• Xonotic, an arena FPS in the style of Quake 3 and Unreal Tournament that has a thriving online scene – including non-standard modes like Defrag
• Hedgewars, an artillery game that’s basically “Worms: Armageddon”, but with some annoyances removed (like stray solid pixels get automatically removed, jumping is easier to do, etc.)
• Teeworlds, a fast-paced 2D multiplayer action game
• The Battle for Wesnoth, a turn-based strategy that (unusually for a PC title) adopts a gameplay structure more reminiscent of console TBSs (comparable to Advance Wars or Fire Emblem, but not a clone of either)
• Minetest, an attempt to make a Minecraft clone that has since turned into a very flexible engine for block-based games

And personal suggestions:

• Mindustry and shapez.io have already been mentioned by another person here, they’re worth checking out
• Shattered Pixel Dungeon is a newbie-friendly roguelike
• speaking of roguelikes, NetHack is a classic, although not as user-friendly and way more difficult
• Simon Tatham’s Puzzles, a collection of small games that all implement classic puzzles (Solo is sudoku, Pattern is “Japanese crosswords” aka Picross, etc.)
• LAN Master (video, download), a legit public-domain open-source homebrew NES game where your goal is to connect all computers into a network by rotating pieces – obviously you’ll need an NES emulator for that, FCEUX, Mesen, Mednafen, ares and Retroarch are good FOSS choices for that