An important part, changing the original Constitution's 3/5 of a person for slaves, is now changed to "the whole number of persons". This concretized making people of color being fully counted for purposes of "representation", meaning redrawing maps for purposes of assigning how many Congressmen a given district was authorized.

More importantly, former slaves (property) are made FULL CITIZENS of The United States and of the State in which they reside.

But to you point, what I think your question is getting at, Federalism now is Supreme over State's Rights. Civil war's rallying cry was State's Rights. The Union, having won the war, is now telling the States that they are no longer completely Sovereign and should forever forget such dreams. "You States are now merely cells of a larger body."

This is the final Victory of Federalism. Remember, just after the Revolutionary War, a weak Federal Government was created. The Articles of Confederation was too weak. So there was much debate (hence the Federalist Papers, Thomas Paine) trying to persuade the States of the necessity of having an actual Federal Government.

After the Civil War, the victory of the Federalists was complete.

The part of the 14th Amendment which most affects us today is "No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States". What this does is apply things like the Bill of Rights (first 10 Amendments) to the States.

Marbury vs Madison: "Constitutions are superior documents".

So in conclusion, 14th Amendment means that State laws must be in accordance with the Federal Constitution. They are subject to Federal review in the courts. So, for examole, if someone gets caught with a pistol in his car, this has to do with the 2nd Amendment. It is a "major Federal question" which allows that case, in theory, to go all the way to the U.S. Supreme Court. (if the State law seems to violate the 2nd Amendment)

Don't even get me started about California's gun laws. That's a whole different rant