A game grabbing alot of attention lately is Terraria. It is similar to Minecraft in the fact you can mine, and explore. Levels are randomized which allows tons of replayability.
Another 2d game with lots of random maps is Diablo 2. Although, I think that classifies as a 2.5d because it is isometric.
Plants vs Zombies, Angry Birds, and any other games on the net fall under 2d are usually going to be on the same line of quality.
Civilization III is 2d / isometric, and I would spend hours a day on that game.
As for games on the line of Jazz Jack Rabbit, all I can think of is Lost Vikings. I believe Blizzard has made it capable of running with out a DOS emulator. A modern 2d game is Death and a Fly, you can look into that.
An older, but cool 2d shooter with online capability is Soldat. There is a 2d Half life which is also cool.
AOE ( Age of Empires ) is kind of 2D and there is plants vs zombies, Angry birds like dbonetrain said. Also there is Chronotron - which isn't old but still good. You can also download flash games, those are 2D and still are fun.