There are a bunch. This looks very low-tech, but is completely free and kinda fun for a while:

http://www.airline-empires.com/index.php?/page/home.html

This is much more slick:

http://www.airlinesim.aero/en

But none come close to the complexity of the real thing, and the models are unrealistic/can fairly easily be "hacked" (print money by giving away flights but charging $1000 for meals, etc).

I haven't found one with the right mix of fun and complexity. They're all either super fluffy with five minutes worth of gameplay, or spreadsheet managers like EVE (without even the pretty graphics to accompany it).

Same thing for airport simulators.

I can’t remember the name at all but there was a very interesting web-based airline sim we ran when I was in school for aviation. There weren’t any fancy graphics or anything though. You just bought airplanes, set routes and prices, and managed your finances. A fiscal quarter would run every few days. It was a lot of fun!