The term 'Fortean' comes from Charles Fort, a writer and researcher, who spent several decades in the early 1900's collecting and studying reports of events that lay beyond the accepted beliefs of the time.
It is quite a large umbrella term and many things fall under it – ghosts and hauntings, lights in the sky, anomalous creatures, psychic abilities, extreme archaeology, magic, strange natural weather phenomena and much, much more besides, including missing/disappearing people.
Charles Fort, for whom the term is coined, spent many decades in the early 1900's searching through old journals, newspapers, reports, magazines and other media for stories and reports of strange events. He then researched those findings further, and made case studies which he would collate and have published in books.
Fort never really attempted to prove the events he wrote about, but rather stated the facts as known on the matter. He once stated he never believed anything he wrote about, but that never stopped him being interested and still looking further.
Before Charles Fort died he had amassed quite a number of followers of his work, and without these early adopters of his work, the various paranormal fields and communities could be quite different. It is not entirely certain what Fort died from, as he did not trust medical science and doctors, and refused to seek medical help. He held out long enough to see an initial run of his famous (non-posthumous) book "Wild Talents".
For many it is used to describe such events as:
Falling frogs and raining fish – at the time quite the mystery, but today many accept explanations that the fish, frogs and other animals/creatures have been plucked out of their habitats with freak weather events and deposited over nearby towns and cities.
Archaeological finds that fall outside the realm of our common understandings, such as living frogs found trapped in stone, non-natural artefacts that are said to be aged in the billions of years old, advanced technologies for the times.
Spontaneous human combustion, the sudden eruption and burning of a human with no outside source of ignition.
These and many other types of phenomena are not satisfactory explained by science to all, and in many cases not proven to even exist.