Can I publish my Master's Thesis that I've claimed copyright on? I graduated this past May with my master's in a physical science. Since then, my thesis has been uploaded online by my university, with my copyright on the thesis. I was interested in publishing the results from my thesis in a scientific journal, but with this copyright, is that even possible now? What are the rules when it comes to this sort of thing?
Generally, the agreement that you sign with a journal publisher is not just to allow publication, but rather to transfer ownership of the copyright to the publisher.  After that point you don't own the copyright and can't publish the material your self except under the terms of the copyright transfer agreement.  Some agreements are quite liberal while others are more strict.  

However, most journal publishers specifically allow for the publication of papers that are substantially similar to parts of "published" MS theses and PhD dissertations- under this system you can still make copies of your thesis or dissertation freely available online.  Furthermore many publishers are OK with theses or dissertations that basically consist of chapters that were previously published as papers in their journals.  A few publishers (for example I believe that this was an issue with the American Chemical Society although they've updated their policy) have not allowed these practices.

In short, copyright is most likely not going to be an issue in publishing research papers based on your thesis. 

Another issue that you didn't discuss in your question is rewriting the thesis material to fit into the short form of a journal article.  Thesis chapters usually need to be significantly shortened for journal publication.