The Army is a federal agency, the National Guard is a dual state/federal agency, and the State Guard is a state agency. They play slightly different roles.
Army
The Army is a key component of the U.S. Armed Forces, providing expeditionary land forces wherever and whenever they are required.
Army troops are more likely to be called first when federal interests are at stake (such as a war or for defense).
Army troops are under command of the United States Army Forces Command.
Army troops are prohibited from being used as domestic law enforcement under the Posse Comitatus Act.
Army Reserve
This is the reserve force for the United States Army, and can be called in times of emergency by the U.S. government to serve in the Army.
Army Reserve troops are under command of the United States Army Reserves Command.
It has a similar mission as the Army, except these are reserve troops.
Army Reserve troops are prohibited from being used as domestic law enforcement under the Posse Comitatus Act.
National Guard
Each state has their own National Guard, and technically it is the state’s military, not the federal military. The National Guard can be called by the Governor of the State, or by the President of the United States.
National Guard troops are dual enrolled in the state military and the federal military as reserves.
Under the “Total Force” concept, National Guard units are considered to be Reserve Components of the United States Army.
National Guard troops are more likely to be called by the state when the state’s interests are at stake (such as natural disasters, defense against invasion of that state, search & rescue, etc.). The federal government also has the ability to call the National Guard into service. Typically whoever calls the Guard first, gets control of the Guard (i.e. U.S. President vs. State Governor).
These forces are under the jurisdiction of the State Governor unless they are federalized by order of the President of the United States.
National Guard troops, when under the command of the State Governor, can be used as law enforcement in that state or adjacent state (if welcomed by that state). The Posse Comitatus Act only applies if the troops are federalized.
State Guard
Many states, like Texas, have a State Guard that is similar to the National Guard, the main difference being that the State Guard can only be called by the State Governor, and cannot be called by the President. These troops are designed to support the National Guard, and to defend the state in case the National Guard is called away.
State Guardsmen often assist during times of state emergency, such as during natural disasters or search & rescue operations, and play a supporting role for the National Guard. If the National Guard is federalized and called away, the State Guard takes over their duties.
These forces are under the jurisdiction of the State Governor.
As these are not federal troops, the Posse Comitatus Act does not apply and the state may use state troops for law enforcement purposes if it’s desired (such as restoring the peace during a riot).