What power does the government have on education?

Education is left up to states, I know, but the federal government does play into the basic education standards.

What law/act/place in our government system dictates this right to the federal government?

Civil Rights Act of 1964:

Congress passed an act to desegregate all public schools. Some states opposed, but the executive branch ordered federal troops to enforce desegregation.

The federal government passes laws to ensure basically equal balance of education in all states for the good of the whole union. Many times it's just new programs set up and if a state doesn't follow the program they won't receive an amount of federal $.