Google Voice is a service that provides you with a single phone number that people can call and the call will automatically be routed to multiple phones at the same time, such as office, cell and home. The service used to be called GrandCentral, before Google bought it in 2007. The service has other neat features such free transcriptions of your voice mails so that you can view them like e-mails and cheap international calls (just like Skype).
You can already use Google Voice with any cell phone, but Google has recently released mobile apps for phones like the Blackberry and the Android. The advantage of using these apps to make calls is that the person you are calling sees your Google phone number on caller ID, not your cell phone number, plus these apps include nice features for managing voice mail, sending and receiving free text messages, etc.
Several developers had created Google Voice client apps for the iPhone, and Google itself created the definitive app—only to see Apple reject it. A Google spokesperson revealed to the New York Times and other media outlets that Apple rejected the app several weeks ago, and Apple has now removed from the App Store all of the other Google Voice client apps, such as GV Mobile.
John Gruber writes on his site Daring Fireball that sources have told him that it was AT&T that pressured Apple to remove the apps because it was a threat to AT&T's business model. You would still use AT&T minutes on a Google Voice app, but you could avoid text messaging fees, surcharges on international calls, and certain other AT&T sources of revenue.
Of course, as I recently wrote in my review of the Skype iPhone app, you can also use Skype to make essentially free international calls. But the New York Times says that the Skype app is different because it only works over Wi-Fi and doesn't use AT&T's network. Google Voice would use the AT&T network, so presumably AT&T has the final say on whether the app is allowed. It is strange, though, that AT&T hasn't blocked a Google Voice client for the Blackberry, which also runs on the AT&T network.
The internet is full of posts from people complaining about the rejection of Google's Google Voice app. I understand their point; in a perfect world, every app would be loved by all and approved by Apple. As Harry McCracken writes, the iPhone seems like a computer, and we are used to being able to install whatever we want on our own computers. But the reality is that AT&T subsidizes the iPhone so that you can pay only $200 for it instead of $600, and as a result they get to decide what you can do with it on their network. Hopefully the restrictions imposed by Apple or AT&T will never get so severe that people will decide that the restrictions outweigh the advantages of the iPhone, but it is unrealistic to expect no restrictions. (Having said that, I do hope that Apple continues to let the Skype app work on the iPhone because it is great when you are traveling internationally.) In the meantime, you can still use Google Voice with an iPhone or any other cell phone. You don't get all of the fancy features like a caller ID that displays your Google Voice number, but you can still use the mobile web interface to access many Google Voice features using the Safari web browser.
[UPDATE: I should add that I do feel bad for developers who it would seem are not getting sufficient information from Apple. It would be a shame to see good developers walk away from the iPhone because of fiascoes like this. For example, see here and here, both links via Daring Fireball which has been all over this story.]