The animated GIF is an interesting indicator of where networking is at any given time.
Before 1995, animated GIFs were almost unknown to most computer users.
The World Wide Web really exploded into the mainstream around 1995, and for most of the late 90s the web was new and exciting and nobody knew what to do with it yet. For the first time, it was easy to share files around the world, and since all major web browsers at the time supported GIF animations, these gaudy, experimental web sites were full of animated GIFs. They were eye-catching and dynamic, and nearly every site anywhere had a "This website is permanently under construction" label with a little animated stick figure in a caution sign, like you see at construction sites, except moving.
Interestingly enough, by about '97 or so, the average home computer had become powerful enough to do simple 3D modeling, and 3D animated GIFs were suddenly everywhere as college students experimented with them. Most of them just rotated around over and over again.
By the time 2000 came around, the old 90s style websites looked hopelessly outdated, and animated GIFs were inextricably associated with the old design. GIF animations were suddenly nowhere to be seen as designers sought to distance themselves from the old Web 1.0, Web 2.0 was right around the corner and even though most people didn't know what it was, they knew it was better.
While that was happening, Adobe Flash was getting ready to revolutionize the way we view video clips on line. In 2005, YouTube was launched using Flash technology, and for the first time it was actually easy to view video on line. Before that, a person hoping to view a video on the internet would need to find out what format the video was in and download or install the right software, codec, or browser plugin to watch it. With YouTube and Adobe Flash, that was all embedded right in the web page and all you had to do was press "play" (assuming you had Flash installed, which by then everyone did).
Animations on the web were suddenly popular again, although admittedly they were nowhere near as gaudy and distracting as they were in the '90s. Online shopping sites sometimes used an understated sparkle or gleam to draw your attention to special deals, and Web forums (a product of Web 2.0 design philosophy), which had been using graphical smileys to replace the :-) smileys everyone was familiar with, started using cartoonish animations for some of them, but quickly realized when enough was enough. Going overboard just annoyed everyone.
As the average internet connection speed increased, it became practical to create larger and more complex animated GIFs that would still download and play quickly (even in real time). Today, most animated GIFs are either cartoonish smileys used on web forums, or actual video clips translated to GIF format (losing the sound in the process, of course). The size of the GIF could be kept under control by limiting the framerate or the pixel dimensions of the file, and impressive video could be created for under 10MB (which would have been an obscene amount of data to download just a few years prior).
Translating video to GIF is popular enough today that many internet memes now appear in that format, typically with subtitles (equally likely to be the actual subtitles or some kind of joke) to make up for the lack of sound. Due to its simplicity to create and widely supported structure, creations that once would have been in comic strip or sequential art format are sometimes in animated GIF format now, depending on how tech savvy the artist is and whether it would add anything to the art (punchlines, twist endings, and reveals can no longer be spoiled if your eyes wander to the end of the strip, for example).
Considering the severe limitations of the animated GIF, it's amazing the format has stuck around for so long. Static GIF files are slowly dying out for three reasons. First, the LZW compression algorithm is under copyright, second, they are limited to 256 colors, resulting in unattractive dithering, and third, the PNG format is now available and also widely supported as a viable replacement. PNG does not, however, support animations (more on this below), so it can't compete on that front.
Interestingly, by far the most popular feature of animated GIFs, the ability to loop them multiple or infinite times, is not part of the GIF 89a specification at all. This was part of an addition made by Netscape called the Netscape Application Block, which is so widely supported it doesn't even matter that it's not officially part of it.
Where animated GIFs will go from here is anyone's guess. As internet speeds increase, computer processing gets more powerful, artists push the limits of what users are willing to download, and web designers continue to experiment with what their users will tolerate, they're likely to stay part of the Web experience for a good long time. The format itself hasn't changed since the '90s, the Web has changed around it.
For now, the animated GIF is a time-tested workhorse of the Web that hasn't changed because it hasn't had to, and so many animations exist now that we'll undoubtedly be supporting the format for decades to come just for backwards compatibility with them.