Don't know where to start? What about that book you read? You thought it was similar to what you write, so check the acknowledgements page. Some authors thank their agent in the book. If an agent liked their work, they might like yours. Next step, further research on the agent.
Of course, you will to go to the agency's website, right? Check out client lists, if available. Take note of submission policies. And read the agent's and/or agency's blog. But don't submit yet. You'll be continuing your research into this agent.
Agent Query is a searchable database of literary agents. But it is also much more. It has info on large and small publishing houses, literary magazines, plus articles such as "When Agents Offer Representation..." or "How to Write a Query." In addition, it has links to many other sites, including some mentioned below.
When checking out an agent, definitely go to Publisher's Marketplace. One cool feature is the site lists who has recently updated their page. Click on browse members, where you can look up a specific name or merely see who is listed. But once you're on an agent's page, you can often see what projects they've recently sold and/or best known projects.
Don't miss Chuck Sambuchino's blog* - he's the editor for the Guide to Literary Agents. I love how he's posted a variety of agents' pet peeves about Chapter 1. He also has entries on "How I Got My Agent" plus has good links to other blogs and websites.
Now that you've done some research, you may know who an agent's clients are, so read some books that agent represented. It'll help you learn whether the agent could be interested in what you write.
Know an author agented by this agency? Ask her questions. You may want to find out whether the author has had other agents previously. He may not want to say who he left, but he probably is willing to discuss problems.
Here are two good articles on Harold Underdown's site regarding agents:
• Children's Book Agents and Artist's Representatives: a Guide
(do I need an agent, what do agents do) has good content, but remember it is a dated article as one of the examples is of The Firebrand which has closed.
• Finding and Choosing Literary Agents
A lot of working checking into an agent? Yes, but your submissions will be much better targeted than many an agent receives.
*I know there are many other people blogging about agents. If you'd like to list any, feel free to comment here.