What is difference between a Republic and a Democracy?

"The United States shall guarantee to every state in this union a republican form of government..."
- United States Constitution, Article IV, Section 4

So what's the difference between a Republic and a Democracy?

Democracy:
Invloves the government ruling and making laws for the "greater good" of all people, they may abolish personal rights in doing so. Democracy is government by and for the people. They may or may not be republics--that is, government limited by constitution or charter. The tricky part of "democracy" is defining "the people" and then deciding what counts as "by the people" and what counts as "for the people." In a sense, that could be considered the content of democratic practice.

Republic:
Invloves the government using and abiding by the constitution heavily. Personnal rights are respected and cannot be taken away. This helps to avoid tyranny and mobocracy (the majority makes laws and governs by passion, prejudice, or impulse, without restraint or regard to consequences).
Republics are the common and "standard" type of governments found today, not democracies, despite what many people (who may not know the definition of either) think.
Just as democracies may or may not be republics, republics may or may not be democracies.

The difference between Democracy and Republic:
Democracy and Republic are two forms of government which are distinguished by their treatment of the Minority, and the Individual, by the Majority. In a Democracy, the Majority has unlimited power over the Minority. This system of government does not provide a legal safeguard of the rights of the Individual and the Minority. It has been referred to as "Majority over Man".
In a Republic, the Majority is Limited and constrained by a written Constitution which protects the rights of the Individual and the Minority. The purpose of a Republic form of government is to control the Majority and to protect the God-given, inalienable rights and liberty of the Individual.

The United States of America is a Republic.

"Wherever the real power in a Government lies, there is the danger of oppression. In our Governments the real power lies in the majority of the community, and the invasion of private rights is chiefly to be apprehended, not from acts of Government contrary to the sense of its constituents, but from acts in which the Government is the mere instrument of the major number of the Constituents." -- James Madison, in a letter to Thomas Jefferson, 1788

"The unity of a nation's spirit and will are worth far more than the freedom of the spirit and will of an individual; and that the higher interests involved in the life of the whole must here set the limits and lay down the duties of the interests of the individual." -Adolph Hitler