I thought the US was a Republic and not a Democracy

Am I missing something? I heard Obama making statements about supporting democracy. I thought the US was a Republic? There is a difference? I copied the definitions from dictionary.com and it states that the US is a Democracy even in the definition. What's up?

re·pub·lic

   [ri-puhb-lik] Show IPA
noun
1.
a state in which the supreme power rests in the body of citizens entitled to vote and is exercised by representatives chosen directly or indirectly by them.
2.
any body of persons viewed as a commonwealth.
3.
a state in which the head of government is not a monarch or other hereditary head of state.
4.
( initial capital letter ) any of the five periods of republican government in France. Compare First Republic, Second Republic, Third Republic, Fourth Republic, Fifth Republic.
5.
( initial capital letter, italics ) a philosophical dialogue (4th century b.c.) by Plato dealing with the composition and structure of the ideal state.

de·moc·ra·cy

   [dih-mok-ruh-see] Show IPA
noun, plural -cies.
1.
government by the people; a form of government in which the supreme power is vested in the people and exercised directly by them or by their elected agents under a free electoral system.
2.
a state having such a form of government: The United States and Canada are democracies.
3.
a state of society characterized by formal equality of rights and privileges.
4.
political or social equality; democratic spirit.
5.
the common people of a community as distinguished from any privileged class; the common people with respect to their political power.

Wikipedia says a Republic is this:

A republic is a country with a specific form of government, in which the people, or some significant portion of them, have supreme control over the government and where offices of state are not granted primarily based upon family, military, or business connections.[1][2] In modern times, a common simplified definition of a republic is a country where the head of state is not a monarch.[3][4] The word republic is derived from the Latin phrase res publica, which can be translated as "a public affair", and often used to describe a state using this form of government.

Both modern and ancient republics vary widely in their ideology and composition. In classical and medieval times the archetype of all republics was the Roman Republic, which referred to Rome in between the period when it had kings, and the periods when it had emperors. The Italian medieval and Renaissance political tradition today referred to as "civic humanism" is sometimes considered to derive directly from Roman republicans such as Sallust and Tacitus. However, Greek-influenced Roman authors, such as Polybius and Cicero, sometimes also used the term as a translation for the Greek politeia which could mean regime generally, but could also be applied to certain specific types of regime which did not exactly correspond to that of the Roman Republic. An example of this is Sparta, which had two kings but was not considered a normal monarchy as it also had ephors representing the common people. Republics were not equated with classical democracies such as Athens, but had a democratic aspect to them.[5]

In modern republics such as the United States and India, the executive is legitimized both by a constitution and by popular suffrage. Montesquieu included both democracies, where all the people have a share in rule, and aristocracies or oligarchies, where only some of the people rule, as republican forms of government.[6]

Most often a republic is a sovereign country, but there are also subnational entities that are referred to as republics, or which have governments that are described as "republican" in nature. For instance, Article IV of the Constitution of the United States "guarantee to every State in this Union a Republican form of Government".[7] The Soviet Union was a single state composed of distinct and nominally sovereign Soviet Socialist Republics.

Which is it?