Yes, with the arrival of Mountain Lion on the Mac today, Apple has finally extended its Game Center iOS gaming social service to the desktop. The easiest way to find Game Center on your Mountain Lion-equipped Mac is to jump on over to the Launchpad, where Apple's already set up an icon for you. Click that sucker, and you'll be able to sign in, either with a brand new account or (even better) the same account you use on iOS.
Once logged in, you're able to see all of your friends, any leaderboard ranks and achievements you've racked up on iOS games, and any pending friend requests you happen to have. Unfortunately, there aren't a lot of Game Center-enabled games on the Mac yet, but a few popular games have been able to make the jump, including Cut the Rope, SpellTower, Osmos, and Firemint's popular Real Racing 2. I'm sure we'll see plenty of other games making the jump soon enough -- part of Game Center's charm is that it's relatively easy for developers to implement and include in their titles.
Other than that bigger screen (the wood and felt look really great, actually), and the fact that desktop games will be included, this implementation of Game Center isn't radically different from what we've known before. There are still achievements and leaderboards, and presumably there will be multiplayer tie-ins soon as well. But perhaps it's enough that Apple has enough faith in its social gaming system to finally bring it back to the desktop, and let it grow there, too.