Amazon allows you to self-publish Kindle books (you don't even need a separate account - scroll down Amazon's homepage, click on the "self publish with us" link and sign in with your normal account details).

Amazon allow you to self-publish whatever you want - fiction, non fiction, short books of 3000 words, and massive books of 100,000 words. And they pay well too - you get 30% of the taking if the book price is between $0.99 and $2.99 and 70% if the price is above. Rule of thumb is that short books go for $0.99 and longer books go for more - you should be able to establish the pricing by looking at other books in your niche.
But first you need to write your book. Make sure it is unique and original. Choose a subject that you are an expert about, it really comes through in the writing. Take some real trouble with this, your success depends on producing something of genuine value. Perhaps you love cooking and have a lore of fine recipes that you've created. Or perhaps you have information to help someone who has been in a difficult situation. And if you are a fiction writer, here's a chance to write that novel you have always had in you.
Once you've got it written, get someone to proof read it and edit it (try to get someone who will give you real feedback - family are better than friends for this in my experience as they arn't afraid to say what they think).
Next you need to upload your book. First save it as a word document (a .doc file). Before you submit to Amazon, I recommend that you submit to [Smashwords](https://www.smashwords.com/about/supportfaq). Smashwords is stricter about what they accept, and only accept original works. Best of all, they issue free ISBNs. This will help you no end when you come to submit to Amazon.
Read the Smashwords style guide before you proceed - they like you to have a proper copyright notice at the start of your book, and everything spell-checked and proofed. Once you are ready, all you need to publish your book on Smashwords is your word document, and they do the rest, producing pdf files, kindle files, files that are compatible with the Apple store and Barnes and Noble. They will then distribute your book to Apple and Barnes and Noble, as well as allow you to earn direct sales from the Smashwords site.
Once you have published on Smashwords and obtained your ISBN, you now need to submit to Amazon.
First convert your Word doc to html (simply re-save it as an HTML file). Next, download [Mobipocket software](http://www.mobipocket.com/en/downloadsoft/productdetailscreator.asp) which is free. Mobipocket are owned by Amazon. Click on "create a blank page" and upload the html file and any html image files associated with your book.
Mobipocket will then create a prc file and this is what you upload to Amazon. When setting up your Amazon page, add your ISBN, confirm to them that you are the sole owner of the material, upload your bookcover (which you can have professionally made or make yourself in Paint). and then publish. They will review the material and within 48 hours your book will be live in the Kindle store.
Of course true success comes with promoting your book. Amazon have a kindle publishers community where you can interact with other authors and get advice on how to promote your book.