Amendment X to the Constitution of the United States of America states:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people.
First of all, we must keep in mind that governments (federal, state, local) do not have
rights; they have
powers. Only people have rights. The Tenth Amendment defines a class of powers and then defines their status.
"... not delegated to the United States by the Constitution,..." - This first clause of the definition excludes all powers that are specifically delegated to the United States, referring to the federal government. Most of these powers are described in the original text of the Constitution, some are described in later amendments.
"... nor prohibited by it to the states,..." - The second clause excludes all powers that the Constitution specifically says the state governments do not have. For the most part, these are listed in Article I, Section 10, which states:
Section 10. No state shall enter into any treaty, alliance, or confederation; grant letters of marque and reprisal; coin money; emit bills of credit; make anything but gold and silver coin a tender in payment of debts; pass any bill of attainder, ex post facto law, or law impairing the obligation of contracts, or grant any title of nobility.
No state shall, without the consent of the Congress, lay any imposts or duties on imports or exports, except what may be absolutely necessary for executing it's inspection laws: and the net produce of all duties and imposts, laid by any state on imports or exports, shall be for the use of the treasury of the United States; and all such laws shall be subject to the revision and control of the Congress.
No state shall, without the consent of Congress, lay any duty of tonnage, keep troops, or ships of war in time of peace, enter into any agreement or compact with another state, or with a foreign power, or engage in war, unless actually invaded, or in such imminent danger as will not admit of delay.
The final clause, "... are reserved to the states respectively, or to the people," designates the status of these powers. They are reserved to the states or the people. The federal government does not have these powers. Period. For example, until the Sixteenth Amendment was ratified in 1913 (and there is question as to whether it was properly ratified), the federal government could not assess an income tax. But the states could, and many still do. Should an amendment be added that says states can't do that, then that would become one of those powers prohibited by the Constitution to the states.
A strict interpretation of the Tenth Amendment leads to the conclusion that a great number of the powers the federal government now exercises are unconstitutional. The amendment essentially says to the federal government, "You can do these things and no more." Unless it's something specifically listed in the Constitution, the federal government shouldn't be doing it.
This, of course, is one of the reasons the amendment process was incorporated into the Constitution in the first place. The authors knew that they couldn't plan for all eventualities to come so they created a mechanism whereby the Constitution could be modified to deal with things they missed, or with things that would develop in the future. But they also made sure that amendments could not be passed easily. The Constitution is a "living" document in this sense but it grows and evolves slowly, not subject to the whim of the moment.