While the Constitution gives a lot of power to the national government, the states still have their own local levels of government. Federalism is the system of how power is shared between these levels of government (not to be confused with the Federalists).
Exclusive powers are those allocated for the national government, such as declaring war, while reserved powers are those allocated for the states, as per the Tenth Amendment, such as policing. The powers shared between the national and state governments are concurrent powers, such as taxation.
There are many ways in which power is distributed between the national and state governments, also known as fiscal federalism. Revenue sharing is essentially just national funding to the states with no restrictions on the usage. States love these!
Block grants are national funding to the states with minimal restrictions. Categorical grants are federal funding to the states with specific limitations attached. For instance, a state may receive a categorical grant and have to use the funding on a specific thing outlined by the national government.
Mandates are simply requirements that the states must follow. Sometimes the national government will help fund these mandates. Sometimes they are unfunded, such as the No Child Left Behind Act.