If you had a Sega Genesis / Mega Drive, you probably played Disney's Aladdin, which was released back in 1993.
Die Hard Trilogy (1996) was one of my favourite games, simply because each movie in the Die Hard series was represented by a different sub-game and a different type of gameplay. With this game, you got a third person shooter, an on-rails-shooter and a driving game (where you raced around New York City in a taxi as Samuel L. Jackson screamed obscenities at you).
Released in 1999, Ape Escape is a PlayStation platform game that gives you control of a character called Spike, whose mission it is to travel through time and capture monkeys.
Army Men 3D was surprisingly fun!
Battletoads was a series of games that were released between 1991 and 1994.
Castlevania: Bloodlines (1994) is a cult classic. You'll often see it being featured on some of the "top games ever" lists that people make.
Clock Tower (1996) was a point-and-click adventure that was released on the PlayStation. Throughout the story, you solved puzzles and investigated building while a mysteriously villain called "Scissorman" stalked and attacked you.
Crash Bandicoot (1996) by Naughty Dog looked extremely good for a title that was released in the mid 1990s!
Dino Crisis saw you take control of a red-haired special agent called Regina, who is sent to investigate a secret military facility that is filled with dinosaurs.
Doom (1993) paved the way for other first person shooters.
I absolutely loved Driver (1999). For me, the best part about it was the ability to edit recorded footage of your car chases.
Earthworm Jim (1994).
Ecco The Dolphin (1992). I always found this one to be extremely difficult.
Final Fantasy VIII (1999).
Golden Axe (1989).
Gran Turismo 2 (1999).
The Legend of Zelda: A Link to the Past (1991).
LMA Manager (1999).
Medal of Honor (1999).
Mega Man X (1993).
Metal Gear Solid (1998). My favourite game of all time. The first time that I played it, I spent days
Mortal Kombat (1992).
NBA Jam (1993).
Oddworld: Abe's Oddysee (1997).
PaRappa the Rapper (1996).
Resident Evil (1996).
Resident Evil 2 (1998).
Resident Evil 3: Nemesis (1999).
The Lion King (1994).
Silent Hill (1999).
Anna Kournikova's Smash Court Tennis (1998).
Sonic the Hedgehog (1991).
Spyro the Dragon (1998).
Streets of Rage (1991).
Syphon Filter (1999).
Tekken 2 (1996).
Tomb Raider (1996).
Tomorrow Never Dies (1999).
Teenage Mutant Ninja Turtles: The Hyperstone Heist (1992).
WCW Nitro (1997).
World Cup 98 (1998).
WWF Attitude (1999).
Super Mario World 2: Yoshi's Island (1995).
FIFA Soccer 96 (1995).
GoldenEye 007 (1997).