(This page covers US Copyright. There are different rules in other countries.)
Copyright shows that you legally own the stuff you create.
The moment you write, draw, sculpt, design, or whatever something original and unique, you own it. It is copyright to you. This paragraph right here? It is copyrighted to me and you can't legally use it unless I give you permission.
When you write a story, it becomes your intellectual property the moment you create it.
You can do whatever you want with the story: copy it, sell it, give it away for free, read it aloud in the city square, print it on wrapping paper, throw it out, etc. No one else has that right unless you give them permission.
Registering Copyright
Even though you own whatever you write the moment you write it, that doesn't mean you can easily defend your copyright in court if it comes down to it. Unfortunately, there are far too many thieves, plagiarists, and pirates out there today who will copy your poem, story, or book and try to sell it or give it away themselves.
If you want to protect your copyright legally, you need to register it with the US Copyright office. It doesn't cost much, and it's an important way to protect the stuff you spent so much time and energy on.
What About Mailing Myself a Copy?
This idea has bounced around forever. You mail or email yourself a copy of your story and don't open the file up to, supposedly, prove that you own the file and its contents. While this might create a useful backup if your computer explodes, it doesn't give you legally enforceable copyright protection. Register your copyright officially and you don't have to worry about it.
Note: Copyright and copywrite are two different words. Copywriting is writing copy... advertising content. If you copywrite, you will own the copyright of that copy.