Here I've collected rules and links for various games I think are worth the time to play.
♘ Board Games
Reversi/Othello - A simple to learn territory control game that's hard to master.
Mill (Nine men's morris) - Classic game where pieces are arranged three in a row to capture an enemy piece or block all of their moves.
Hive - An easy to pick up strategy game. It's board-less (though you will need a surface), using distinct hex shaped pieces interlocked in a fight to capture the opponent's queen.
Love Letter - Board Game Geek's game with the highest rating in ratio to the lowest complexity. Simple game to learn and play; perfect for convincing those family members who don't want to play new games to try something better. Easy to carry anywhere. Has none of the issues normal social deduction style games would have.
Arimaa - A game designed to be simple for humans to learn and play than computers. Can be played using a standard Chess set by using the height of the pieces to determine their power, but is not quite a Chess variant.
🃓 Card Games
Tiến Lên (Thirteen)
🎲 Dice Games
Dice 10,000 - Score climb game with catch-up mechanics.
Ship, Captain, Crew - Push-your-luck score based game playable with 5 six-sided dice.
🕮 Tabletop RPGs
Fate (Free PDFs) - A generic medium weight narrative system that encourages cooperative story building. Simple core rules are expanded to their fullest potential, making it an easy game to remember, hold in your head, and improvise with. Lessons learned playing this game can be applied to any future game, even if you're not exactly using the rules.
Fate (SRD) - Easy to access online collection of rules, and additional rules and tweaks you may want to use.
Mini Six Bare Bones Edition - An improvement on the OpenD6 rules, this game comes in a small package and provides everything you need to play any kind of adventure. Easy to pick up with no need to read hundreds of pages of rules. My personal choice for a more traditional style TRPG. Can easily be tweaked to give players more Fate-like narrative control.
OpenD6 - Still worth a look, and can be used as rule suppliments for Mini Six.