iCloud access through browser
Modern Answer (2016):
This is a feature of iCloud Photo Library, that launched in 2015 (and was in beta in 2014). With the updates to Photo Stream/iCloud Photo Library, your photo library is now viewable on iCloud.com.  

Original Answer (2012):

This feature is called Photo Stream, and at this point, there is no web version of it. 

The previous iteration of iCloud, which was called Mobile Me, did have a gallery option, but did not have the direct upload integration like Photo Stream (you had to manually post images).