In Vampire: The Masquerade, the independent clans are those major clans that claim descent from Caine but are not part of either the Camarilla or Sabbat. The clans were introduced in the first edition and have been expanded since. The four clans are the Giovanni, Followers of Set, Assamite and Ravnos clans. As of the Revised Edition, the Gangrel are independent as well, its elders having formally severed the clan's ties with the Camarilla.
Each clan's reason for independence varies: in the case of the Assamites and Giovanni, there are specific deals with the Camarilla that guarantee autonomy. The Followers of Set just weren't interested, and the Ravnos never participated in any party they couldn't crash first. Each clan also has certain eccentricities that separate them from the mainstream of Cainite culture - the Followers of Set, Ravnos and Assamites all view Cainite vampires with a certain degree of disdain, and the Ravnos and Assamites in the West are effectively tips of an iceberg. There are far more Ravnos and Assamite vampires in their home territory and their structure is more complex there, not least because of a relative lack of Cainite competition in the area. As for the Giovanni, their goals are largely orthogonal to Cainite society and they are just as happy to be left alone. Finally, the Gangrel left the Camarilla over concerns that the sect was ignoring threats to their very existence: specifically, a creature that they believed to be an Antediluvian.
Despite their independence, most of these clans are willing to work with the Camarilla, or at least within their territories, within reason. Most Gangrel retain some ties to the sect before the clan broke away. The Giovanni have seen their business with the Ivory Tower increase since the two worked together to retake the cities of New York and Boston. Finally, a large faction of schismatic Assamites has allied itself with the Camarilla since the Methuselah ur-Shulgi arose and began purging those who did not renounce all faiths other than the Path of Blood.
Note that most bloodlines are also independent, with a few exceptions. Several bloodlines are primarily allied with the Sabbat: the Ahrimanes, Blood Brothers, City Gangrel, Harbingers of Skulls and Kiasyd, as well as various outside clans' and bloodlines' antitribu (including the Country Gangrel, House Goratrix, the Panders, and the Serpents of the Light). The True Brujah, Nagaraja and Old Clan Tzimisce were allied with the Tal'mahe'Ra before that sect disintegrated. While Gargoyles are nominally independent, a number of them remain slaves of the Tremere and are thus affiliated with the Camarilla, if only by association.
The notion of "independent clans" is one that only dates back to the formation of the Camarilla; in the Dark Medieval era before the rise of the modern sects, the only distinction between clans was that between the High and Low Clans.