We live in a republic (a group of states that are self governing under the umbrella of a centralized or federal government). The type of system we employ is a representative form of democracy (meaning the people vote for people to represent their views instead of directly participating in every bit of legislation). This was done to weigh the needs of the people (the house of representatives) in balance with the needs of the states (the senate).
Republic and democracy are descriptions of two different things (one is a form of goverment, the other is the system of goverment). So, for example you have the former Soviet Union, which was a republic, but not a democracy. You also have India, which is a democracy but not a republic. The US is both. N. Korea is neither.
The president is elected through the electoral college system instead of by direct vote so he can be a split between the senate and the house (the states and the population). The electoral college's purpose is one of the most misunderstood in political science.
Republicans have little to do with republic, and the democrats have little to do with democracy. Both parties support a democratic republic.
The main difference between the parties is how to go about their goals. Both parties want the same things, but disagree as to the means. In general, republicans want people and businesses to be free from government restraints, and democrats believe that government is often the best solution for our nations problems.
For example, race relations. Republicans ended slavery, passed the voting rights act, and passed affirmative action. They thus limited the government's ability to impede someone based on skin color. Democrats have pushed for quota systems under affirmative action, dictating to schools and business who should receive admission. They thus used the government as a solution based tool.
The same with economics. Republicans favor lower taxes as a way to limit the government and allow individuals and businesses to better the economy (capitalism). Democrats prefer to increase taxes, because they believe the goverment is better positioned to better the economic prospects of everyone with the revenues (soft socialism).
Last, in the legal sphere republicans believe courts should rule on laws created by the representative branch of government (the house and senate). Democrats believe that the courts exist to serve society and redress wrongs, even if this may sometimes conflict with current laws.
This is an overly simplistic answer to your question, but I hope it helps get you started. (By the way, the US goal in Iraq is to establish a parlimentarian democracy. The country will be made up of provinces instead of republics).
Best of luck to you.