If I didn't register my copyrighted material, and another one did. Do I lose my copyright?
The author, or the copyright holder if the author has sold or assigned the copyright, may register a copyright at any time, even many years after the work was copyrighted.
However, for all modern works (since 1978 in the US) the work has copyright protection as soon as it is created and set down in a tangible form (such as on paper or in a computer file), whether it is registered or not. Registration cannot gain or lose copyright protection. It is evidence of the copyright, and it provides some additional rights if an infringement suit should occur.
If there are multiple co-authors, one author should ideally register on behalf of all of them, listing all co-authors, but multiple registrations will do no major harm.
No one who is not an author or copyright holder of a work (or a valid authorized agent of the author or copyright holder) may legitimately register a copyright in that work.
Registration procedures and the exact benefits of registration vary by country. In fact, some countries do not have any copyright registration.
Edit: In the US, registration procedures are described in Circular #2 from the Copyright Office. Falsely registering a copyright is a crime under 17 U.S.C. § 506(e), and could be reported to law enforcement. See also Wikipedia's article on "Copyfraud". If a false registration has been made, the true author can file a conflicting registration. This should be sufficient to allow a copyright infringement case to be filed. It would be very wise to consult a lawyer with experience in copyright infringement issues before taking steps in the matter.
It is one of the elements of an infringement claim that the plaintiff owns the copyright, or has been authorized by one who does. often this is not disputed, and is passed by with no significant time or effort. But if a false copyright registration has been made, in which a person who was not the author claims to be the author, then evidence clearly establishing the actual authorship will probably be needed if the case goes to court. A registration establishes a rebuttable presumption that the person who registered the copyright owns it, and this presumption would need to be overcome.