Common law is the system of deciding cases that originated in England and which was latter adopted in the U.S.. Common law is based on precedent (legal principles developed in earlier case law) instead of statutory laws. It is the traditional law of an area or region created by judges when deciding individual disputes or cases. Common law changes over time.
The U.S. is a common law country. In all states except Louisiana, which is based on Napoleonic code, the common law of England was adopted as the general law of the state, or varied by statute. Today almost all common law has been enacted into statutes with modern variations by all the states. Broad areas of the law, such as property, contracts and torts are traditionally part of the common law. Because these areas of the law are mostly within the jurisdiction of the states, state courts are the main source of common law. The area of federal common law is primarily limited to federal issues that have not been addressed by a statute.
http://definitions.uslegal.com/c/common-law/The 'Common Law Court Movement', from 1997:
What the Texas appeals court was just beginning to perceive were the beginnings of a movement created by recalcitrant self-proclaimed "sovereign citizens" determined to wrest control of their lives back from all forms of government or authority. Appearing first in isolated spots in Texas and Florida, the notion of "common law courts" soon spread to Kansas and other farm states, then quickly across the nation. The "common law court movement," as it has somewhat clumsily come to be called, now exists in some form in every state in the country. In some states, activity is minimal; in others common law courts are a serious nuisance; in some, they are a plague on the judicial system. Although featured on television shows like "20/20," common law courts did not really breach the public consciousness until the spring of 1996, when FBI agents surrounded a frigid eastern Montana farm to wait out two dozen recalcitrant tax protesters that locals dubbed "freemen." In reality, however, common law adherents had been active for years in different areas across the country. Frustrated county clerks knew of the strange filings made in their offices; puzzled policemen encountered confrontational motorists pulled over for homemade license plates; irritated lawyers discovered that bogus liens had been placed on their property by court opponents. But there was little public awareness or understanding of the movement. The media reported that Oklahoma City bombing suspect Terry Nichols had declared himself a "sovereign citizen," but treated it as a random, bizarre act by a right-wing extremist, not as an action by someone consciously part of an ideological movement.
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Posse ideology, however, places a far different meaning and reliance on common law. Though there are many different strains and theories of Posse common law, a common thread that runs through most of them is that the common law is a separate, parallel legal/judicial system, one independent from and not subordinate to statutory or written law. For example, throughout the 1980s and 1990s, Posse adherents came up with inventions such as "common law trusts" and "common law banks." What these concepts have in common is the notion that the normal written laws governing the establishment of trusts or the regulation of banks do not apply to these institutions, because they are beholden only to the "common law." In other words, the term "common law" was attached to the word "bank" as a (futile) attempt to avoid the law.
Every common law theorist or group has a slightly different explanation for the origins of and nature of their version of "common law," but the following broad summary of their beliefs is general enough to hold for most circumstances. The key, as mentioned above, is that Posse adherents believe in "common law" as independent of (and even hostile to) other alleged legal systems, rather than all being part of a whole (9).
According to common law doctrine, the common law originated in the Middle Ages to protect property rights. The American Revolution destroyed allegiance to the British crown, but kept common law rights of property. This situation made every man "sovereign" over his own property. Neither Congress nor state legislatures nor county or city ordinance nor judicial ruling by any courts could deprive people of their common law rights, including their rights to "allodial" property (an ancient concept describing property that could not be lost for failure to pay taxes; it never applied in the United States, although some states did enact "homestead" laws). Grievances were to be settled by common law juries which decided the facts and the law of the case.
http://www.adl.org/mwd/COMMON.aspA lot more at that second link. Basically, it's a right wing movement who claim they have a 'common law' right to set up their own courts to decide on matters traditionally covered by English common law, and that no statute law can be passed to override that - which they they say means they don't have to pay taxes. They tend to say this while heavily armed. It's fairly closely linked to the militia and Ruby Ridge types.