The first thing you must do to publish your story is to know what kind of story you are trying to promote. Is it book length or a short story? Is it fact or fiction? If it's factual, what is the subject? If it's fiction what is the genre, science fiction, romance or historical? You will need to know also the word count as many publishers have minimums and maximum amounts of words they will accept.
Get a current edition of the "Writer's Market." This book is indispensable when it comes to looking for just the right publisher. Every book and magazine publication is listed with important information such as the contact person's name, tips to break into their market, phone, website and addresses. They also tell you what percentage of "over the transom" pieces they publish, and the percentage of agented pieces they accept. Most importantly they will tell you how they want the story presented to them.
Prepare your story to be sent out. Most publishers ask first for a query letter. In this letter you will tell them who you are, what the story's about, and why you are the best one to write it. Sometimes you will be asked just to send the whole manuscript. If you do that make sure you retain a copy in case it gets lost or ruined. You will most likely be asked to send along an SASE (self addressed stamped envelope) for it's return in case it's rejected.
Understand that the best and most sure way to get your story published is to keep sending it out until you have a publisher. After every couple of rejections you may want to reevaluate the type of publishers you've been sending it to. Perhaps they've already had a similar story, or maybe they don't publish that type of story. It's OK to make changes here and there if you believe it will increase your chances of getting published. Just don't give up.