What most people mistakenly assume about hiring a literary (book) agent, is that they won't have to do anything at all to promote their book once the agent is hired. Completely untrue.
Let's look at the kinds of circumstances when you would or would not need an agent. For the sake of clarity, if you write poetry, short stories, essays, and even most non-fiction, you won't need to hire an agent to sell your work. In the case of the first three, no agent who isn't a con-artist would sell them for you, and in the case of the last, although a reputable agent will typically get you more money, a better contract, and a beefier advance, they are not absolutely necessary for selling your book – even to a big publishing house.
If you've written a novel, or creative fiction, and you want to publish your book with a major publishing house, you will need an agent to broker the contract for you. Your agent, once hired, would find you the very best deal available in terms of the advance, the meat of the contract, and the percentage of profit going in your pocket. Most agents are motivated because if they don't get you the best possible deal, they don't make money. A typical cut for a literary agent is 15% of whatever they get for you, so it makes sense for them to aim high.
Before you approach an agent, make sure that your book is as polished as it can be. That means, it's been proof-read extensively, cut, edited, and offered to family and friends for free in exchange for serious critiquing. And take some of their advice into consideration. After all, they're probably a good cross-section of your target market for readers.
The most important thing is that you understand what your agent does and does not do. An agent works on commission attempting to sell your book to the best possible publishing house and negotiates the terms of your contract. Though they can and do offer guidance and suggestions, can get you an advance, or help you find an editor, they are not there to hold your hand and help you rewrite your novel. And steer clear of any agent that promises you fame and fortune, or offers you guarantees or unrealistic expectations. A good agent will be realistic, supportive, and honest. That means telling you even the hard stuff, especially if they can't sell your book.
Once your book is polished, and you've decided that you need an agent to help you sell it to a big publishing house so that you get a great contract, it's time to start the research phase. The first thing you need to do is find books that are similar to the book you've written and research which agents represent the authors. If this information isn't listed in the author's credits, you can always contact the publishing house and ask them. When you have a list of at least five literary agents who might be interested in representing you, send them a very brief, tailored query letter of no more than one page explaining why you're contacting them, giving a short summary of your book, a very brief biography that focuses on your work as a writer, and a closing that offers your manuscript and explains your end purpose.
If the agent is interested in seeing your manuscript they will either ask for the entire thing or a few chapters. Before mailing it out, and if you haven't already done so in the research phase of finding your agent, you need to go over the submission guidelines for the agency.
This is only one of the ways to find a literary agent. You can use databases, information found on writers' forums, Internet searches, and even word of mouth from other authors to find a great book agent.