There are a lot of different options available for creating an online photo album.
Most people are probably going to go with one of the free online photo storage websites. Most of these are owned by a big player.
Popular free online storage sites are as follows:
If you have your own internet storage available, an option like Google's Picasa can help you store your pictures and create online photo galleries.
Other options are installing open source photo gallery software like Gallery or Coppermine on an internet host. One issue here is that you need to keep the software update to date to ensure there are no security risks on your site, otherwise your site might get hacked. This would be particularly annoying if you were using private password protected galleries.