Once a book is written, how does one go about getting it published?
Congratulations on writing a book, that is a difficult task to accomplish in itself. However, the most difficult part has yet to come. To publish a book, you must prepare a proposal (sort of an outline/summary of your work) which is submitted to book agents who work on your behalf to get the book accepted by the publisher and hopefully published. This is not an easy task and can cost money. Here is an outline of what the process involves. If you don't want to hire an agent to sell your books to big publishers, you can attempt to do it yourself, but again it takes big time, expertise in marketing and selling and then creating a contract about your book. You can read how to publish on your own here. Lastly, you can always publish online, but this is a new model and not your common trend to get the exposure you want. Read more about that here.
What is the genre of the book? In what category does it fall?
I currently have a book sitting with an agent (a fiction novel) so I might be able to offer some additional advice.