An animated graphic is a series of images which when viewed in sequence at a particular speed is perceived by the brain as a single image. Minute differences between the images create the illusion of motion. This is possible for us to achieve because visual information is processed at a fairly consistent rate of roughly 1/25 of a second in the vast majority of human brains. We really cannot visually discern any event that happens faster than that. Thus, if we view a sequence of images at around 30 per second, that means that the image and the transition to the next image occur in the space of 1/30 of a second - which is faster than the brain can "see". Thus we have an illusion of it being a single image.
That is the basis of motion pictures. There are all kinds of varieties and variations, but it pretty much always comes back to that basic principle. There are three elements to that principle. 1. The number of images in the sequence 2. The minute differences between the images 3. The frame rate - or rate at which the images are displayed.
These three elements combine to give us what I call "Transformation Over Time". So I define an animated graphic as one which shows a particular transormation over a particular span of time.
The advent of computers has made it possible for everyone to create animated graphics - whereas before computers this was very complicated, and extremely expensive endeavor.There are numerous applications which create files which show these transformations over time. In fact, you could easily make the case that any application which allows you to work on graphics on a computer screen does this. You could, but I won't.
The kind of files that I'm mostly concerned with are .MOV files, .GIF files, .FLA files and .SWF files. These are just the files that I work with because I work on a Mac. These aren't the only file formats available on Mac or PC, but they are the one that I work with.