In today's market, with today's prices and offerings, to play @ 1920x1080 with High settings and good FPS I would recommend the GTX 470 or ATI 6870.
How do I base this? I play @ 1920x1080. I like all High settings. Personally, 4xAA is fine. My eyesight isn't perfect so more than that isn't a big deal. But with a single GTX 470 you can play Bad Company 2, Just Cause 2, and other such games @1080p with just about everything set to High as long as you don't crank up AA. I generally see 50-60FPS with these two games so long as I keep AA settings down. To me, this is the sweet spot because I have plenty of FPS, and my 'minimum' FPS is almost always above 30FPS. So I never see the "stutter" effect when things explode on screen.
The ATI 6870 is a pretty good competitive match for the GTX 470. So that's why I recommend that as an option with it. Both cards can be found for about $230-$250 for more basic "reference" design models.
Here's just a couple examples of cards you
could buy:
Asus GTX 470 ($269.99 - $20 = $249.99 - $30 MIR =
$219.99
http://www.newegg.com/Product/Product.aspx?Item=N82E16814121372&cm_re=GTX_470-_-14-121-372-_-Product
PowerColor ATI 6870
$229
http://www.newegg.com/Product/Product.aspx?Item=N82E16814131373&cm_re=ati_6870-_-14-131-373-_-Product