I am a young student in California and I have developed a theory, which I believe might affect the scientific world, though, through fear of it being taken or wrongly affected by others, I have kept it to myself.
I am a corporate attorney and have advised clients of protection of the intellectual property integral to their operations. In the intellectual property world, there are three types of federal registration you can pursue - patent, trademark, and copyright. Patents secure ideas, trademarks secure brands, and copyrights secure written product. If you are looking to protect your idea, you may think about filing for a patent. The first step is securing a provisional patent that sort of holds a spot for your idea while your actual patent application is pending or your idea goes into development. The patent process is complicated and lengthy, so I would recommend getting the advice of a patent attorney if that is something you want to explore. Not all ideas are able to be patented, and you may find that a patent is not an option for you. The other option is to maintain your idea as a trade secret. This essentially means that you are careful about who you share it with, and when you do share it with anyone, you have an Non-Disclosure Agreement in place that prohibits them from disclosing your idea to anyone else or misappropriating it for themselves. If you had a company built around the idea, you would establish an in-depth trade secret program designed to prevent disclosure of your idea outside of the company. (e.g. Coca Cola has a trade secret program built around the recipe for Coke.) As an individual, you would just need to be careful and, to be extra thorough, get people with whom you share your idea to sign NDAs. There is no federal registration process involved in maintaining a trade secret, just common sense and consistency with your protective practices.
While Catherine does a great job of explaining your options below (from a business POV) I believe that it would help you and us if you were able to provide a bit more background on what space/area your theory is involved in & what your intentions are. There are many fields of science such as life sciences, behavioral sciences, physics, etc.
To say that you have a theory that you haven't shared with anyone and presumably haven't tested a bunch of times would probably not go over too well within the higher echelons of the scientific community. As an entrepreneur, I constantly tell other people that ideas (or "theories") don't mean much, but rather what you do with them does.
In very simple terms, if you'd like to be the first person to be documented as having this theory then there are things you can do like publish a book/article or even record a video/video blog and email it to yourself or your attorney. Doing this should time stamp your introduction of this information to the known world.
If you want to get real deep then keep reading...
Apparently, there is an interesting "theory" about theories themselves called "The Underdetermination of Scientific Theory" which basically says that a scientific theory can only be disproven and not proven: https://www.quora.com/Why-can-a-scientific-theory-only-be-disproven-and-not-proven
The same user on Quora goes on to say, "In practice, this distinction often turns out to be largely semantic. Theories are not proven, but "accepted", with the notion that we might retract that acceptance in the face of new data. But that's under-playing the fact that a good theory has been tested many, many, many times, and it's quibbling to act is if there's a serious chance that it's incorrect (especially if you're not actively working on a better theory that also explains just how the old theory managed to survive so long)."
You can cheaply file a provisional patent which will last for one year. After that year you have to convert it into a 'full' patent, or it becomes public knowledge. If you're at a university, you can talk to their patenting department, and if they like your idea they'll pay for the full patent process.
If you want advice on the provisional patenting process, or the full patent process I can help you out. I've filed several.