What is a regional constitution?

In the political context, a regional constitution refers to the laws that govern a particular locality within a given country. A good example is the united states. The country has two levels of government each governed by a set of laws. The state and the Federal governments are co-sovereign. This means that while the US has a national constituency,  each state has its constitution that regulates the affairs of the state government. The Federal Constitution is superior while the state one is subordinate. As such, the supreme law of the land can be used to trump any regional law that is inconsistent with its provision, and the local constitution must align with the Federal one. The regional constitution is mainly used to grant the people in that region exclusive rights that the Federal law does not make arrangements.

A regional constitution may refer to the constitution of a certain locality within a larger nation.
For example, the United States operates under a system of dual constitutionalism in which the federal and state governments are co-sovereign powers. While the US has a national constitution, each individual state also has its own state constitution.
The US Constitution provides that federal law is the supreme law of the land, meaning that any federal law can trump a state law that's in direct conflict. However, the US Constitution is a limited grant of enumerated powers, so the federal government cannot pass laws in all areas. The Tenth Amendment to the US Constitution also makes it clear that whatever powers are not granted to the federal government are reserved for the state governments. 
The individual rights granted under the US Constitution cannot be diminished by state constitutions. However, a state constitution can provide that state's residents with more rights than they would have under just the US Constitution. 
Many people do not know that every US state has its own constitution, nor do they know the ramifications of that fact. Ever increasingly, though, citizens are bringing lawsuits based upon their rights under state constitutions.