Writing a research paper means you are bringing your thoughts and opinions into the scholarly conversation on a particular topic and you'll be drawing from the thoughts and ideas that others have already written about the topic. It's important to give credit to those who have already written about topic - this helps your ideas stand out from theirs, and it allows others reading your paper to track back to where you get your ideas. Your citations, both in text and in your reference list let readers trace the words and thoughts you're using back to their original source.
Proper citation and avoidance of plagiarism is one aspect of Academic Honesty. The Georgia State University Policy on Academic Honesty is available online.
Often, a general rule that's stated is "if you're in doubt, cite it", and that can be a good place to start from. Some specific places you must use citations in your research:
Despite the "when and doubt, cite it" phrase that you may here, there are times you don't need to (or shouldn't) cite a work in your research.
For the last two in particular, if you are unsure about whether an item is considered common knowledge or an accepted fact, go ahead and either ask your instructor or include a citation. It's better to be more cautious than less.
Finally, you should also never include citations for works you didn't actually use for your research.