Who's asking: Various clients, at various times
Once again, I'm giving away work for free, but only because this may be the professional question that makes me feel the most helpless and incompetent.
A good agent is advocate, therapist, editor, financial advisor, publicist and coach. The search for one is more difficult than looking for a job, and only slightly less difficult than looking for a spouse. Anyone who says otherwise is either very lucky or trying to sell you something.
That said, I have some feeble advice to offer, and if any of my author pals feel like expanding on the advice, please do.
First, don't approach an agent before your manuscript is finished. If you have three good chapters and an outline, great; finish the book. We've all heard stories about the authors who sold books based on a proposal scribbled on a napkin, or a great first chapter, or a five-minute pitch; we hear about these things because they're freaks. Advances for first novels are rarely enough to let you quit your day job to finish the book, so let that fantasy go.
In the real world, you're wasting your time and the agent's when you submit unfinished work. If an agent rejects the first three chapters of your unfinished work, he or she won't want to look at it again once it's finished. If they like the book and want to represent it, they won't want to wait three months, six months or however long it takes you to finish -- if you do finish.
You can, however, do some homework while you're writing. Professional associations are useful sources of names and contact information, especially in genre fiction. I'm skeptical about the value of pitching to agents at writers' conferences, but it always helps to meet potential agents in person.
Look for books and authors that appeal to the audience you're writing for, and check the acknowledgements. Most authors thank their agents, and if an agent likes a book that resembles yours, they might like your book, too.
Many reference books and websites provide information about the best way to approach individual agents. The book I use is The Writer's Market, but other directories have similar information, and you'll find several in any decent library. Before you send out your query, check the agent's website for updated information about whether they're accepting submissions, and how they prefer to receive them.
And don't put all your eggs in one basket. Identify as many potential agents for your work as possible, and submit your work to all of them. It's not like sending stories out for publication; if more than one expresses interest, terrific.
Finally, don't pay anyone any money up front. Agents earn their living by selling their clients' work; publishers pay advances and royalties to the agents, who take their percentage and send you the rest. Legitimate agents don't ask for money up front, and don't charge "editing fees." (They may suggest that you seek help from an independent editor, and may even recommend one -- I've had clients referred to me this way -- but those are independent arrangements between writer and editor, and the agent receives no compensation for these referrals.)
Finding a literary agent is hard, even for established authors. I know a few successful authors who have had to go through this process more than once, because their original agent retired, or because the relationship just wasn't working out.
If it was easy, everyone would be Dan Brown (and then Dan Brown wouldn't be Dan Brown...).