To this day I don't understand how anyone can stand to use or view photos on Flickr. To me it is the most terribly designed UI of any photo sharing site. Its popularity amazes me. Even Facebook, etc. seem better. That being said I probably don't know how to use it right, but it sure aint intuitive.
Anyway, for free options, I personally think Picasa is the best. Not as full-featured as Flickr I'll grant, but photo viewing is a lot smoother and nicer (IMHO), it has nifty face recognition, links nicely with Picasa, and it has a reasonable (though simple) permissions system. If space is an issue upgrades on space are *very* cheap (Google just lowered the prices):
# 20 GB - $5/yr
# 80 GB - $20/yr
# 200 GB - $50/yr
# 400 GB - $100/yr
Not too bad at all.
If you're willing to pay and want something more fully featured with probably the nicest gallery presentation around, check out SmugMug
. Its categorization system is a little wonky, though it does include arbitrary tagging capability. Beyond that I'm very happy with it. Non-profits get a free "Pro" account, which is very nice. I only hesitate to ever recommend it because it's not free and there are many other free options. But the features are pretty nice IMO...