The X Amendment reads thus: "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." Clearly, then, where the Constitution is silent, the States are reserved broad powers. They cannot become dictatorial--their constitutions must not conflict with the national supremacy of the US Constitution--but they can have broad latitude in determining statewide regulations, taxes, and the like.
In theory, at least, this federalist structure is how our nation is supposed to operate, and it manages to do so, despite significant hobbling from the federal government. Congress has forced upon the States a number of unfunded federal mandates. Essentially, a large portion of State budgets are consumed with fulfilling orders from Washington, D.C., without any form of assistance. Additionally, States are often coerced into adopting certain policies or passing certain laws, lest the federal government withdraw their funding (this tactic was used to increase the drinking age from 18 to 21--not necessarily a bad thing, but the means matter almost as much as the ends; such coercion circumvents the proper amendment process).
What brought about this change, and how can we reverse it? How can we restore the proper balance between the States and the federal government?
There are no easy answers here, and the centralization of power in the federal government occurred for a complicated host of reasons: the acceptance of a desperate people of a greater role for the government in the economy during the Great Depression; the (temporary) success of a massively planned economy during the Second World War; the massive expansion of the welfare state during the Great Society; the (necessary) fight at the national level to protect the civil rights of black Americans; and more.
However, I would argue that a major source of this problem was the passage of the XVII Amendment.
The XVII Amendment replaced the old system of selecting senators with their direct election. Prior to its passage, senators were selected by their state legislatures, which were themselves chosen in local elections.
There are a number of compelling arguments for why this amendment was adopted. For one, many states had already moved to a de facto system of direct election, in which voters essentially "elected" their senator, and the state legislatures were duly pledged to vote in accord with the people's choice. Also, there were several scandals in which senate candidates merely bribed state legislators for their votes. Finally, many state legislators found that voters cared more about who the legislators would elect to the Senate, not what they thought about state problems.
You can review these arguments in a (rather condescending) piece from Slate by David Schleicher entitled "States' Wrongs."
"[T]he States no longer have a constitutional role in the federal government."
However, while there certainly appeared to be need for reform in senatorial elections, many of these problems still persist. Voters are still overly-fixated on national politics, even more so than voters in the late nineteenth and early twentieth centuries. If anything, state elections are even more focused on national issues than they were before. Special interest groups still manage to exert influence over the Senate, and can do so even more effectively by whipping up voters.
Most importantly, though, is that the States no longer have a constitutional role in the federal government. The Senate used to serve as the representative of the States' interests, while the House still operates as the representative of the people's interests. Now the people have direct influence over both branches of Congress, and an important, necessary brake on the fickle will of the majority is gone.
States' rights has become an ugly phrase, associated as it is with slavery and segregation. However, just because states' rights has been invoked to defend the indefensible doesn't mean that it isn't a good idea. The States function as an important bulwark against tyranny, and federalism affords many opportunities for policy innovation and experimentation--Louis Brandeis's "laboratories of democracy." Also, the geographical, ethnic, religious, and cultural diversity of the United States practically demands states' rights, as different States have different needs, goals, and desires.
Repeal of the XVII Amendment is extremely difficult and unlikely: people like to vote (actually, people like to know they can vote, even if they often choose not to do so). But Congress, specifically the Senate, can do much to keep the further expansion of federal power in check. Senator Ben Sasse of Nebraska is spearheading this effort through his speeches, delivered from the Senate floor, about the proper role of the Senate and its obligation to be an august, contemplative chamber.
We, the people, can also take steps to become more involved in state politics. Ultimately, the drive to restore federalism starts with us.
For more information about the XVII Amendment and different approaches to addressing it, here are some resources:
The Campaign to Restore Federalism (pro-repeal of the XVII Amendment): http://www.restorefederalism.org/
"Repeal the 17th: Problems to Address" by constitutional scholar Rob Natelson: http://tenthamendmentcenter.com/2013/08/26/repeal-the-17th-problems-to-address/
"Repeal the 17th Amendment?" by Gene Healy of the Cato Institute (great piece that is sympathetic to the idea, but recognizes the political problems involved): http://www.cato.org/publications/commentary/repeal-17th-amendment
"States' Wrongs" (mentioned above) by David Schleicher of Slate (anti-repeal, with some interesting historical background and a lot of elitist sneering at movement conservatives): http://www.slate.com/articles/news_and_politics/jurisprudence/2014/02/conservatives_17th_amendment_repeal_effort_why_their_plan_will_backfire.html