Government

A parliamentary system (whether it's a republic or a monarchy) is a system of government where the executive branch is not entirely separate from the legislative branch. The legislature, or parliament, typically gets to choose from among its members who becomes the head of the executive branch, or "head of government"- this head of government is normally the leader of the political party that holds the most seats in the parliament. The legislative branch wields significant power over the executive, and can even pass a special law, often called a "vote of confidence", to fire the head of government. Countries that use a parliamentary system include Germany, the United Kingdom (and most of the Commonwealth countries), and most countries of Eastern Europe.

The opposite kind of system is the presidential system, where the head of government (who is also usually the "head of state"- essentially the symbolic leader of the entire country, similar to the monarch in a monarchy) is separate from the legislature, and operates with much less constraint from it (and it's much more difficult for the legislature to fire the head of government, if they can at all!). Countries using the presidential system include the USA and pretty much all of Latin America. Some countries, like France and Russia, have a so-called semi-presidential system, where they have a president separate from the parliament who is a powerful head of state, but then they also have a prime minister who is chosen by the parliament. In France, it's set up so that if they disagree on how to run things (as happened in the late 1980's when the President came from one party and the Prime Minster came from an opposing party), the President handles foreign policy while the Prime Minister handles domestic issues.

A democracy is any style of government where the people play a role in running the government; there is no dictator, no monarch, and so forth. "The people" are said to rule. There are two primary kinds of democracy- direct, where the people directly run the government (as in the ancient Greek city-states, where all free, adult men would gather together and decide the laws); and representative, where the people choose (vote) for someone to run the government for them (as in the United States, Germany, etc).

The opposite of democracy is despotism, where one person- a "despot"- has unlimited power to rule as he sees fit; the people in such a system have very little (if any) power to do anything without the despot's consent. A dictatorship is a very similar, almost interchangeable concept. A despot is usually a monarch- a king, an emperor, etc. A dictator is generally not a monarch of any kind, and is usually a military leader (like Napoleon- although he eventually became a monarch), a political leader (like Hitler) or something else along those lines. Otherwise, they're virtually identical.

So, to answer your question, they are two separate, but not mutually exclusive, systems for running a particular government. A country can use one or the other, or, as in the case of Germany, both.