There are some good resources you can link to right here on enotes to find information about government. Also, see the U.S. government website below for information on individual states. The White House site has very specific information on how the states are governed.
Briefly, the United States, as you know, is a federal system. That means we have a central government AND individual state governments, hence our name, "The United States of America." This idea was conceived by our founding fathers as a way of dealing with the great diversity in a country as large as the United States. Under the federal system, the states have a certain degree of autonomy and sovereignty with regard to their local state governments, but they are not allowed to do anything that would supersede or jeopardize the federal government or go against the Constitution of the United States. Historically, this has caused many conflicts over states' rights vs federal government, the largest of which, of course, was the U.S. Civil War. In this case, the southern states believed they had certain rights regarding trade and slavery and that if they didn't like what the federal government was doing, they could pull out of the union. The federal government declared secession illegal, and it is still considered as such today. All powers not delegated to the U.S. government nor prohibited to the states are retained by the states according to the 10th amendment of the U.S. Constitution.
Four of our U.S. states are established as what is called "Commonwealths" - KY, MA, PA and VA. This technically means that they are based on the common consent of the people - hence the term, "common" wealth. An American is a citizen of both the U.S. and his/her state, and state citizenship is flexible, so Americans are free to come and go freely or live in whatever state they choose, unless they are a convict, and that's another story.
States have different laws than the federal government and usually they are referred to as "state statutes." Those laws govern things such as education in the individual states, real estate, taxes, commerce, public safety, etc., as they apply to the state. So, some states, for example, tax peoples' retirement pensions and some do not. Some states have no state income tax, but most do. All states except Nebraska are set up like the federal government in that they have a bicameral Legislature, two chambers (House, and Senate). Nebraska has a unicameral state government - only one chamber.
Over the years, the federal government has taken on a larger and larger role in governance, sometimes to the chagrin of the individual states, and there have been many lawsuits between the states and federal government - some very famous cases in Supreme Court history. However, the individual state constitutions affect the way the states govern within each individual state. This is why you see on the news today that in certain states, they are approving same sex marriage, or mercy killing, etc., and there is debate as to whether an individual state can or should be able to pass such laws, especially when they might go against the federal laws. Right now, some states are thinking of suing the federal government over health care, believing the federal government does not have the right to force citizens to purchase certain health care coverage.