This thread might succeed better if it were just "Mario games" or something of the like, might lead to more discussion which is what we want.
I love the Mario franchise, I played the games as a kid and I have very fond memories of doing so. My personal favorites are the following:
1. Mario Kart Wii - This is my favorite Mario series of all time, I grew up playing Mario Kart 64, Mario Kart DS, Mario Kart Wii, and Mario Kart 7. I've wasted hours on these games and they are some of the best games I've ever played overall. Despite being released over a decade ago, Mario Kart Wii is still played a ton with the creation of CTGP and Wiimmfi for online play. This series will always remain a favorite for me.
2. Super Mario Galaxy - I loved the Super Mario Galaxy series, completing different, unique levels with the objective to find stars and save the planet was a fun concept for kid me. The original version was my favorite, I played the game through over and over multiple times and somehow never got sick of it.
3. Mario Bros - Yet another good series in the franchise, and the most classic one. There are tons of different versions of this game, overall it's just a classic and a great series.
Those are my top favorites, though I'd like to try Super Mario Maker sometime if I ever get around to it. The rest of them haven't really appealed to me as much.