Flickr allows one to upload images into their own library. They can choose to make these images public or keep them private. They can also include metadata -- data about the data -- like a description of the image, the geocode (latitude and longitude) of the place it was taken, and user-defined tags for future retrieval.
If one chooses to make an image public, it will be discoverable by search engines like Google and Yahoo. It will also be discoverable by Flickr service search. This makes it available on the public Internet.
What is one to do if they have thousands of images on Flickr? There are two mechanisms to help organize and retrieve images.
The first is that they may be included in "sets" of images which can in turn be organized into "collections." These are analogous to directories and subdirectories on your hard drive.
The second organization and retrieval mechanism is tags. You can assign one or more tags to each image. These tags are simply words that come to mind when you see the image, for example, you might tag a picture of your dog pet, animal and rover. The image would then be found in a search on any of those tags.
Publicly shared images can be found by anyone on the Internet. One can also create a Flickr group for sharing images among a small group of people who share a common interest. This might be a family, students taking a class, people working on a project, etc.
An interesting aside -- Flickr was developed by a small company building a multi-player Internet game. The game company was failing financially, so they decided to focus on just its capability to share information among players. They reprogrammed that, turned off the game service (to the dismay of their game customers) and turned on the image sharing service. They were subsequently acquired by Yahoo. You can hear an account of this history by company co founder Caterina Fake.
Here is a short Flickr tutorial.
See this Economist article on Flickr and tagging.