Under the 10th Amendment, powers not enumerated by the Constitution, such as requiring drivers license, defining and creating rules for marriage, creating laws regarding abortion, creating and maintaining an educational system, or powers not specifically enumerated in the Constitution are left to the states or the people to decide.
Federalism was an ingenious plan for a couple reasons.
1. It gave the Federal government just enough power to establish and maintain a stable infrastructure, such as creating and maintaining a postal service, a military, to regulate commerce to create a stable economic environment, etc.
2. It prevented the Federal government from making laws that would infringe upon state and natural rights. In essence, it told Congress what it could rule upon (enumerated powers) but what it could not rule upon (10th amendment).
3. The founding fathers thought Federalism was a good way to prevent the federal government from passing laws that created risky programs that might bankrupt the nation. They knew it was okay for a state to create risk because the federal government would be able to bail out the state. Yet if the federal government took a significant risk and failed (and Obamacare is a great risk), protections created by the Constitution would likewise fail. It is for this reason only states are allowed to create laws regulating abortion, healthcare, education, etc. If one state does well, others may copy. Yet if one state fails, others won't copy. That's a major advantage of Federalism, allowing sort of a trial and error system within the states. Yet if the federal government fails, who will bail us out? The founders thought of this, and that's why they created a system of Federalism.
Bottom line: Federalism was a system to limit the scope and power of the Federal government, and to protect state and individual rights. It was meant to prevent the federal government from getting out of control.