At its most basic, the term Paganism, is an umbrella term covering all religions that are not related to the Judeo-Christian family of religions. The word itself once denoted those “country folk” who were not up on, or didn’t care about, the fashions of the day, including the state religion. Since Judeo-Christianity has been the dominant, indeed, only acceptable religious choice for some time, it has come to mean those who aren’t Judeo-Christian. This would include all “Primitive” animist and polytheistic religions that are common among native groups as well as the Classical religions of ancient Greece and Rome (though it wouldn't have in their heyday). It does not include Hinduism or Buddhism because those religions are dominant in their native areas and socially acceptable choices elsewhere. It would also exclude all religions that have stemmed from Judeo-Christianity, including Satanism and various syncretic religions.
The Druid author and scholar Isaac Bonewits subdivides Paganism into PaleoPaganism, MesoPaganism and NeoPaganism. When most people say the word “Pagan” in reference to modern day religion, they really mean “Neo-Pagan”. Neo-paganism is really a subset of the larger term “Pagan” and is itself an umbrella term covering a wide range of beliefs.
There are many definitions of Paganism. I'd be interested in hearing what everyone else has to say.