Enumerated powers are those powers given to the federal government that are specifically mentioned in the U.S. Constitution. For example, Article I, Section 8 grants the U.S. Congress certain powers such as coining money, regulating interstate trade and commerce, declaring war, raising an army and navy and to establish laws of immigration.
Many enumerated rights are also listed in the Bill of Rights. The reason there is a Bill of Rights is that a majority of the colonies were unwilling to accept the Constitution (to ratify it) without a specific, enumerated list of rights which they feared would not be respected if they weren't specifically included within the Constitution. So, the first 10 amendments were added and the Constitution was ratified by the colonies.
According to the 10th amendment, all powers not enumerated by the Constitution are left to the states to decide. In this way, establishing rules for marriage, abortion, driving cars, education, Internet, etc. were left to the states to decide.
This system is called federalism. It's where two governments rule over the same geographic area. The Constitution establishes these two governments as the federal government and the states. The federal government (meaning Congress and the president) can only pass laws regarding the enumerated rights. If it is not an enumerated right, it is left to the states to decide.
This system was essential, because many of the states had religious schools, and they did not want the federal government telling them they could not have these. There were many other differences between the states, and federalism allowed each state to maintain its own sovereignty, or the right and power to govern themselves without intrusion by the federal government.
Essentially, enumerated are those powers granted to the federal government. These are the powers Congress can make laws upon, and the president can sign into law. Any laws made by the federal government that are not based on enumerated powers granted by the Constitution should be made null and void by the U.S. Supreme Court.