According to ComScore (and via TechCrunch), Facebook Photos is running away from the nearest competitors: Photobucket, Flickr and Picasa (in that order).
Facebook has been the space's leader for over a year, but the gap has widened and really started to open in September 2008. TechCrunch suggests that it is due to the profile redesign:
"But the tagging feature has been part of Facebook Photos for a long time. What happened in September to accelerate growth? That is when a Facebook redesign went into effect which added a Photos tab on everyone’s personal homepage."
How about two other additions to why the marked growth is occurring (although I totally agree that the timing is due to the profile redesign):
1. The iPhone App is one of the most popular and addictive. And it makes photo taking / sharing dead easy. In fact, it is easier to upload a photo than to enter your 'status update.' Based on my network's feed - photo usage via the App has grown significantly.
2. Switching cost: once you begin to upload photos onto Facebook, it becomes difficult to move them elsewhere or start uploading elsewhere. The switching costs are high and there are network effects. As Facebook grows (and it is everywhere now), there is less of a reason to share via print sites like oFoto / Kodak, Picasa, Shutterfly and so forth.