Super Mario Bros.
The revolution of video games basically started here with the success of the NES and Super Mario Bros. You play as Mario (player 1) and Luigi (player 2) and go through 8 worlds (3 levels each with a castle level) to fight Bowser and save Toad and Princess Peach.
Super Mario World
Now we had to the SNES. Guess what? You are now in Dinosaur Land and starting off at Yoshi's Island to figure out Bowser has kidnapped Princess Peach. You play as Mario (player 1) and Luigi (player 2) and they even introduced Yoshi. Yoshi makes it easier to move from the levels faster but sadly you cannot take him with you into castles.
Super Mario 64
Mario gets an invite from Peach to come to her castle. When you get there you figured out Bowser kidnapped the princess. (seems to be a pattern here) This is the first 3D Mario game that launched with the N64 and it even gave you more to do in levels beside getting to the end. You can only play as Mario (DS remake allows you to take control of Yoshi, Luigi, and Wario) but that's why it is called Super Mario 64. It also introduces shell-surfing and Metal Mario (Metal Wario) and a bunch of new hat combos.
Super Mario Sunshine
Mario and Peach are going on vacation but their days in the sun gets cut short when Shadow Mario gets Mario thrown in prison. Now Mario is in charged with cleaning up the entire Island with a trusty talking water pump and guess what Shadow Mario is Bowser's son, Bowser Jr. and he kidnaps Peach. Now Mario (with the aid of Yoshi and his water pump) must stop Bowser and Bowser Jr. A very easy last boss and again Sunshine added on to Mario giving him a water pump that everyone likes.
Sonic the Hedgehog
The first Sonic game where Sonic must stop Eggman from taking over the world. The special stages allow you to collect the 6 Chaos Emeralds (7 from Sonic 2 and up) in order to fully beat the game. Sonic the Hedgehog also introudced fighting the last boss at the end of every level. A first thing in game history (considering Bowser from SMB wasn't really a boss for all you did was jump over him)
Sonic Adventure
Dreamcast was very successful at first with Sonic Adventure (but PS2 forced them out). Sonic Adventure introduced E-102, Big the Cat, and Chaos. Sonic must collect the 7 Chaos Emeralds before Eggman to stop him from making Chaos into Perfect Chaos and destroying Station Square. First Sonic game that has the characters going through a different stages.
Shadow the Hedgehog
(I consider Sonic Heroes to be one of my lest favorite Sonic games) The first time since Knuckles Chaotix (Which didn't sell well) that a Sonic character besides Sonic had his own game. Shadow must find the secrets of his past by collecting the 7 Chaos Emeralds and either working with or defeating Black Doom. Shadow the Hedgehog also introduces the use of guns, swords, and vehicles and level choices (good, bad, neutral). Plus with the hero and villian gauge you can either use Chaos Control or Chaos Blast.
Sonic Rush
The DS promised us a good Sonic 2D game and they didn't go back on that promise. Sonic Rush is by far one of the best Sonic games. Sonic Rush has you able to control Sonic the Hedgehog or Blaze the Cat. Sonic is heading down Leaf Forest enjoying the day but he is confronted by Eggman. After defeating Eggman a Sol Emerald appears but before Sonic can claim it Blaze appears and takes off with it. Sonic must go through special stages (much like Sonic 2's special stages) to collect all 7 Chaos Emeralds while Blaze has to defeat the bosses. The levels are fun and the music is good and the tension bar allows Sonic/Blaze to use a speed-up move that allows them to plow through enemies with ease. I have done time trial (which once you done it gives you a ghost of that character) and Sonic is faster then Blaze so Shadow is still the only person to equal Sonic in speed.