In order to copyright lyrics, all one needs to do is fill out the relevant forms at the applicable copyright office and ensure that all documents get processed. Which forms are required will depend primarily on where one is living and what form the lyrics are in. It is important to note that it is only necessary to formally copyright lyrics if one is worried about copyright infringement, because most areas recognize the rights of a lyricist over lyrics from the instant they are created in a fixed form. If one's lyrics are primarily kept private but one must send them to others for one reason or another, it may be a good idea to register the copyright to protect one's self just in case anyone attempts to act dishonorably.
When a person makes a creative work like lyrics, that person has the right to copy those lyrics and may do with them what he or she wishes. The primary constraint is that the lyrics must exist in a fixed form. This means that the lyrics must be written on paper, sung on a recording, or documented in some other way. It is not possible to copyright lyrics that one is thinking through in one's head or that one sang out loud without any evidence to support that claim. One can still revise the lyrics and have rights to the revised versions, but unless they have been documented in some way there is no way to copyright lyrics.
If a person wishes to formally copyright lyrics, the process typically involves filling out forms and sending the relevant documents away to a copyright office. Sometimes the entire process can be completed online. Since the fees associated with copyright processing are typically per project, it may be wise to consolidate all the lyrics one wishes to copyright into one document. Depending on whether the lyrics have music, one will either want to fill out a form that applies to text or a form that applies to sound. Even if the lyrics are sung, the words themselves are still protected by copyright.
This process can take several months, but once lyrics have been sent off, one has evidence that one's lyrics were created at a certain point. Some people claim that mailing a copy of the lyrics to one's self is equivalent to copyright protection, but this is untrue. There is no substitute for registered copyright protection, and although one has rights as the creator of lyrics, the legal protection offered by registration is without substitute. In the unhappy situation that one needs to go to court over lyrics, a formally registered copyright can be the best evidence possible.