Muzzy Lane Software's Making History II has all the trappings of a modern strategy game: overlays, maps, troops, production cycles, macroeconomics, the whole business; but you can play it right off of IE or Firefox. ML is going for a Facebook-style re-imagining of the strategy genre: you and your friends can start a game, keep it running in the background (or even close the browser), and receive email/text message alerts for when it's your turn. Think of it as the modern version of the old chess-by-mail games that our grandfathers (or great-grandfathers) used to play.
Just because MH: II runs off a browser doesn't mean that it's light on features: now, i'll be the first one to tell you that it's not a Crysis-style graphical breakthrough, but there are more than enough gameplay features to make up for a lack of visual polish. For example: you can play as literally any nation in the world at that time. You can play Turkey or Iraq if you like, you're just not going to have the most exciting game in history. You have to keep track of political unrest in each of your regions, apply principles of macroeconomics (go ahead and build arms if you can, but don't flood the market or your factories are going to sit idle), protect vital supply lines and conduct battles, all at the same time.
The best part? MH: II isn't staggeringly intimidating. I won't name names, but I've tried to play a strategy game or two in my time and been completely overwhelmed and turned off by unreadable menu on top of confusing menu, sandwiched in between other impossibly confounding menus. MH: II's GUI is simple, efficient, and intuitive.
This is a game I can see getting into with a couple friends of mine, taking a turn or so a day, and slowly dominating the world. Check out the demo for free at Muzzy Lane's website.