When you paraphrase something, you are using your own words to restate the meaning of an existing quote or piece of text. Paraphrasing is a critical skill to develop in research and expository writing, or you risk plagiarizing (which will earn you an F in most high school and college classes).
Sometimes, a paraphrase seeks to expand upon or clarify the original text. For example, the phrase "the lettuce was brown" could be paraphrased as "the lettuce was too old to use in the salad." Other times, a paraphrase is just a simple restatement and recasting of existing text. But again, be careful here: you want to be sure you're not a plagiarist, so learn to paraphrase the right way. You can't just take an existing text and rearrange a few words or use a thesaurus to replace a couple of words and consider it a legitimate paraphrase. This is still plagiarizing. You must change both the language and structure of the original text, and you should add new material to make your words support your own thesis or suit the purpose of your own document.
Most research papers will have a mixture of paraphrasing, original writing, and direct quotes. Of course, any time you are using a direct quote from any source, use quote marks and always cite your sources.
For more help on paraphrasing, look to Purdue University's Online Writing Lab (The Purdue OWL). This website offers some steps in becoming an expert paraphraser as well as exercises to improve your paraphrasing skills (and thereby make your writing stronger).