Enumerated powers are powers given to Congress and clearly mentioned and defined in the U.S. Constitution. Implied powers means that, while not specifically mentioned or defined, it may be inferred by the remaining document.
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.Here are listed three natural rights: life, liberty, and pursuit of happiness. There are many other natural rights, although they are merely implied in this document. Since natural rights was a common theme at the time, the founding fathers figured they didn't need to list them all because they "assumed" people would know the rest.
However, most states were not willing to sign on to (ratify) the constitution without a specific enumerated list of rights and an enumerated list of powers granted to the federal government. It was for this reason that the Bill of Rights was created. They feared that without these 10 amendments state and individual rights, which are implied by the Declaration of Independence and Constitution, might 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.
However, there was still no way that the founding fathers could list every enumerated power, and so many continued to be implied. When Congress makes a law regarding an implied power, this may result in a conflict between the states and the federal government. The resolution of such conflicts is the responsibility of the Supreme Court, which has the sole purpose of making sure all laws passed by Congress are Constitutional.
A perfect example of an early disagreement between the states and the federal government regarding an implied power occurred during the presidency of George Washington, when Washington used the power of the Executive Branch for something that was not specifically enumerated in the Constitution.
Alexander Hamilton wrote a bill that made it through both branches of Congress. The bill was opposed by Thomas Jefferson on the grounds that the Bank of the United States was not authorised by the Constitution. Washington agreed, but he sided with Hamilton's argument in favor of the bank.
The argument used by Washington was that, since an enumerated power granted Congress the right to collect taxes, it was reasonable to "assume" that Congress could also create a bank to hold those taxes and act as a bank for the government.
The most recent example Congress and the president using implied powers was the passage of Obamacare. While republicans used the 10th amendment, the Commerce Clause and the fact that the federal government can't force you to buy something as an argument in support of the unconstitutionality of Obamacare, the Supreme Court ruled the implication was that it was a tax and therefore constitutional.
So even though healthcare is not covered under an implied power, and even though the 10th amendment reserves the right to govern on healthcare to the states, the Supreme Court decided it was covered as an "implied power" of Congress.