Picasa is a free image viewer and organizer that has a built in photo-sharing service. It has been owned by Google since 2004.
Picasa has organizational features such as tagging, geotagging, and facial recognition. Basic photo editing functions such as red-eye reduction and cropping are also available. Slideshows are possible, as well as physical printing services.
Picasa Web Albums (PWA) is the photo sharing side of Picasa, and is similar in functionality to Flickr. Users with Google accounts can store and share up to 1 gigabyte of photos.
Though use of Picasa is free, Google Ads will be seen. By using Picasa, you allow Google use of any photos you upload to display elsewhere on the Picasa site, and for promoting other Google services, in perpetuity.
Users can increase the amount of storage and decrease the amount of ads by paying a yearly fee. Current prices are $5 per year for 20 GB all the way up to $4,096 a year for 16 terabytes.
Many Internet connected Blu-ray players and TVs have the ability to access Picasa.
Other similar services include Flicr, DeviantART, ImageShack, Shutterfly, and Snapfish.
Wikipedia has a history of Picasa, as well as a list of all current free and paid photo services.