Many people understand that directly copying text into your paper is wrong, but there are many forms that copying takes. If you were to copy a paragraph, but change some of the words, it might not register on a plagiarism checker, but it's still plagiarism. Likewise, simply using the ideas of another person is plagiarism. You need to inject your own original thought to the piece, even though you may draw from other sources.
Borrowing heavily---or directly copying---from a piece that you previously wrote is self-plagiarism. Though you undoubtedly give yourself permission to do this, it goes against the idea that all of your work should be original. An example of this would be taking a paper that you wrote for one class and reworking it slightly to fit the topic of another class assignment. You can, however, use your research to approach the same topic from a slightly different angle.
It may seem that if you properly cite the facts and ideas in your work that you haven't plagiarized. This isn't necessarily the case. If your work contains no original ideas---even though everything in it is well-cited---it is still plagiarized.
Though harder to detect, it's also plagiarism if you steal another writer's style. This does not refer to choice of words or the use of long or short sentences---you would naturally want to emulate the style of your favorite writers. In this type of plagiarism, you would take another writer's reasoning style and simply inject your own examples into the piece, eventually coming to the same conclusion that the original writer did.