Always a sub-culture, micronations have blossomed in the internet era
I first learned about micronations back in the late 1990s, while reading Neal Stephenson's runaway science fiction bestseller Cryptonomicon. The novel introduces the concept of stateless data haven, one of which was coincidentally created on Sealand the year after Cryptonomicon was released. It seemed like the book was coming to life!
Sealand is one of many micronations, part of a long history of secession from the rest of the world. Sealand is based on an abandoned off-shore WWII anti-aircraft platform in international waters in the North Sea. For a nation of five people, Sealand has a surprisingly engrossing history which includes not only the aforementioned data haven but also currency scares, international weapons infractions, and an international scandal involving Sealand passports being involved in high-profile crimes.
Anyone can declare a micronation at any time. The key difference between a micronation and a fantasy nation, political exercise like a school's "Model United Nation," or role-playing group is that a micronation is tied to a specific physical asset, be it an island, a home, or a parcel of property. Reading this article, you can draw a line around your desk and declare it your own micronation. And many people have done exactly this, for a variety of reasons.
Some micronations seem to be strictly works of whimsy. Others are a form of political protest, seceding from what is seen as a heavy-handed government. Still others seem to be the work of people who may well be suffering from a mental illness, be it autism or schizophrenia or possibly both. And then you have the "art project" micronations, such as those which are established specifically for the purpose of writing about micronations themselves. (Very meta!)
The tricky thing about declaring your own micronation is getting recognized by other (one hesitates to say "real") nations. But short of that, micronations typically involve setting up a charter, a coat of arms, creating a flag, setting up a website (as a diplomatic interface), designating a royal family, and constructing physical signage. Some micronations have gone as far as to issue their own currency and stamps, which are hot collector's items.
If you have ever been curious about micronations, Wikipedia is an excellent place to start. It not only has a long list of all known micronations, it also has separate articles on each one, filled with all kinds of history and fascinating information.