Public law is a theory of law governing the relationship between individuals (citizens, companies) and the state. Under this theory, constitutional law, administrative law and criminal law are sub-divisions of public law. This theory is at odds with the concept of Constitutional law, which requires all laws to be specifically enabled, and thereby sub-divisions, of a Constitution.
Generally speaking, private law is the area of law in a society that affects the relationships between individuals or groups without the intervention of the state or government. In many cases the public/private law distinction is confounded by laws that regulate private relations while having been passed by legislative enactment. In some cases these public statutes are known as laws of public order, as private individuals do not have the right to break them and any attempt to circumvent such laws is void as against public policy.
In the "English" provinces of Canada, the term private law is also known as English Common Law, or just common law. These are judge-made laws. Public Law is that law which is passed by either the provincial legislatures or by the federal Parliament. In Quebec, private law is basically the civil code of Quebec, considered to be the primary source of private law. These laws are interpreted by judges, but this within the ambit of the codal provisions that have been enunciated by the legislators.
All U.S. states except Louisiana share a heritage with English law, so the private law of most U.S. states is called the common law; indeed, the U.S. Constitution makes specific reference to the common law of England as the fundamental law of the federal United States. Some states, such as New York (owing to its Dutch heritage), have strong civil-law influences, and have enacted laws relating to obligations; e.g., the General Obligations Law and the General Business Law. The distinction between what is public and what is private in the law is often a hazy one. Many consumer protection laws are of a public law nature, which limits the ability of companies to engage in transactions that fail to respect the rights of consumers. Most laws that impose criminal penalties are considered public law, as these laws are intended to protect all members of society, and this not just in the areas of interaction covered by contract and tort laws. The legal system of Louisiana follows the French legal tradition, especially the Code Napoleon, except where those traditions are effectively overridden by contrary English-based legal principles in the U.S. Constitution.
Acts of Congress are designated as either "Public Laws", relating to the general public, or "Private Laws", relating to specific individuals. Since 1957 all Acts of Congress have been designated as "Public Law X-Y" or "Private Law X-Y", where X is the number of the Congress and Y is a number sequentially assigned to each act.
So, Acts are signed into Public Law or Private Law. 
In German-language legal literature, there is an extensive discussion distinguishing public law from private law. A variety of theories have been used:
Under the latter theory, both a contract of situation such as the government buying office supplies are subject to private law. There are areas of law which are mixed under that definition, such as employment law, parts of which are public law (e.g. the activities of an employment inspectorate when investigating workplace safety etc.) and parts of which are private law (e.g. the employment contract).
The differentiation is primarily an academic debate, important mainly for the delineation between the competences of the court system and administrative authorities. A statute will normally include a section stating who is in charge of enforcement.
However, under the Austrian constitution the distinction is of some importance, as private law is among the exclusive compentences of federal legislation, whereas public law is partly a matter of state legislation. As a practical result, the distinction is thus a matter of how the constitution is to be interpreted most accurately.