Self publishing is sometimes called indie publishing (not to be mistaken for the independent or “indie” publishers out there). It's not an easy thing to do, but the rewards can sometimes be worth it if you're willing to put the work into it.
You've written a story. Now you want to publish it. The first step is getting it edited. This is the first tricky part of the process. You can have beta readers pick it apart for you, but it would be better if you can find a professional editor to go through it too. That costs money.
Once that's done, you need a cover. You can do that yourself. If you're an artist, you can create it in your style. If you're not, there are quite a number of free stock photos – careful not to use anything copyrighted and always credit where you got the photo – that you can manipulate into a cover. Make sure it doesn't look sloppy. People will judge your book by its cover. If you want a professional cover? That costs money.
Then you're ready to publish. You'll want to check the guidelines of everywhere you're going to self publish. There are numerous places you could go to do this. There's Smashwords, Draft2Digital, and Kindle Direct Publishing. That's if you want to stay strictly e-book. If you want a paperback that's a whole different process.
Another thing to consider with self publishing is an ISBN for your book. Some places to self publish require you to have one. You can buy an individual one or you can buy them in bulk. But there's a good chance at some point you will need one. You can purchase ISBNs through ISBN.org.
There are many great references out there for helping you along the self publishing route. You can check out blogs posts, news articles, and web pages. One such site is Self-Publishing Roundtable. They have a number of interesting blog posts, a podcast, and videos to help you out.
Self publishing is a difficult business. Make no mistake it is a business, and one you have to devote a lot of time to. There are times when the rewards will outweigh the energy put into it as you sell your first book.