There are no "Democratic" ideals, nor are there "Republican" ones. The main philosophies of the two parties are generally liberal (left) and conservative (right), but there are both conservatives and liberals in each party. What we normally hear about the political spectrum is that there is a "left" and a "right." Anywhere in between is called "centrist". The fact is that the political "left" and "right" that we hear so much about only concerns government actions towards economics and "personal" issues. There is more to government, and life, than that.
There are many other political philosophies. If you only get your information from the main-stream media and government education (AKA "public" schools and universities), you may think there are only two viable philosophies: conservatism and liberalism. There are many others. Of note, there is libertarianism (individual rights and responsibilities) and statistism (big government) that cover the same left-right spectrum, but also add the specific role of government in society. There is also democracy (mob rule), socialism (income redistribution, government ownership of business or "far left"), communism (government ownership of everything, including government control of the economy, or "extreme left"), and fascism (private ownership but government control of business and the economy, which is still on the "left") to name a few. There is also a 'laissez faire' or largely unregulated economics in the "far right" column. Let's not forget authoritarianism (theocracy (religious rule), absolute monarchy, and dictatorship, or "extreme right") and anarchy (chaos, no government) as well. Most all have been tried, and all have largely failed over other, more representative types of government.
Libertarianism, as a philosophy, believes in the primacy of the individual -- personal rights; personal responsibilities; personal property; small and limited government. That's the simplicity of libertarianism. In this philosophy, the government exists only to protect individuals from external harm (foreign invasion), and protect individuals from other individuals in cases of fraud and deceit. Individuals are free to interact peacefully as they wish, with little government involvement or regulation of those consensual, peaceful interactions. You have the right to life, liberty, and the pursuit of happiness (where have I heard that before?) as long as your pursuit doesn't interfere with the rights of others to do the same. Both the Conservatives and Liberals want to control aspects of your life. The Libertarians want you to be responsible for your own life.
While libertarianism allows for individual rights, individuals are also responsible for the results of these consensual interactions. We are all products of our individual decisions. Decisions used to have consequences. Now, government has imposed itself to assist those who make bad decisions. Instead of learning from the negative consequences of bad decisions, they can continue to make bad decisions without negative consequences. People become "victims" of their own mistakes. Or they are "disadvantaged" because they continue to make them. Libertarians would rather be allowed to succeed or fail on their own. Charity is an individual virtue, not a government program.
Finally, libertarianism is more than a political affiliation. It is a philosophy and a way of life. We are all libertarian by birth. It has to be trained out of you, which the current educational system in the U.S. is designed to do. Libertarianism is human nature. Libertarians believe that the fruits of our labor belong to us, to retain or give away at our desire. We all believe that we are the masters of ourselves. Libertarians believe that interactions between consenting, peaceful individuals need no governance, oversight, or regulation. We should all believe that we have no right to someone else's property or services.
I believe government is the opposite of liberty. While there is a need for government in society, the government should be limited in scope and responsive to society -- not an overlord of society.
So where do you fall in the political spectrum? It might surprise you. As a start, I suggest you take five minutes and try the “World’s Smallest Political Quiz” at http://www.theadvocates.org/quizp/index.html. There are only 10 questions, and, if you answer honestly to them all, the chart will show you where you really are politically.