What are the current government systems in the United States of America?

There are a number of ways to answer this question, depending on what you mean by a government system.

One answer is to say that the United States as a whole has a republican form of government.  In addition, each of the 50 states that make up the United States each has its own republican government.  A republican government is one in which the people elect representatives to make laws for them.  Because it would be practically impossible to have everyone participate in the lawmaking process, we vote for representatives who do that work for us. 

Another answer is to say that the United States as a whole has a federal system of government whereas states tend to have unitary systems.  A unitary system of government is one in which the central government has all the power and only gives whatever powers it wishes to the lower levels of government.  States are like this as state governments only give city and county governments whatever powers they wish to give.  By contrast, the US as a whole has a federal system where some powers are given to the national government and other powers are given to the state governments.  Neither level of government can take powers away from the other level.

Finally, you can say that America has a presidential system of government.  (The states also have such a system, though their chief executives are called governors).  In a presidential system, there is a separation of powers between the legislative and the executive branches.  (In the US and the 50 separate states, there is also a judicial branch.)  The chief executive is elected separately from the legislature.  This is in contrast to a parliamentary system of government in which the legislative branch elects the chief executive from among its members.

All of these are systems of government that exist in the United States as a whole and/or in  its 50 states.