Things You'll Need
The first, and most important step by far, is to know the market you're going to be pitching. Listen to your local public radio station during "Morning Edition" and "All Things Considered" to get a good sense of the sort of essays they air. Note the length, tone, subject matter, style, even the stories that air before the essays. Editors use essays to complement a timely story or event, or occassionally just to fill a hole on a slow news day. If you can't listen live, check out npr.org, click on "Opinion," then on "Commentary' to find examples of recent essays.
Once you feel like you've got a sense of what NPR editors might be looking for, you can begin writing. But as you do, keep in mind certain news pegs that might make your essay more appealing to editors. Got a story about your son's tour in Iraq? Might be good for the anniversary of the invasion in March. Be sure to get it submitted at least a month in advance of the relevant date (March 19th, in this case).
While writing, remember a conversational tone and brevity are key. After all, you'll end up reading this in your own voice if selected, so you'll want to be comfortable doing so. And as far as length goes, you'll be lucky to get much more than 3 minutes, so keep it simple and pack a punch with as little as possible.
When you feel like you've got a winning essay, it's time to pitch it to the editors. Instructions and updated addresses can be found at http://www.npr.org/about/pitch/commentary.html.
The last step is the hardest--and that's being patient and waiting for a response. The hardest part of this is that NPR doesn't often send rejection notes; you're likely to get the silent treatment instead. So if you don't hear anything after a few months, it's probably time to try another essay.
If your essay is accepted, you can expect some editing and coaching on your speaking voice and recording. If you don't hear anything, don't despair. Check out other public radio outlets like "This I Believe," an independent essay series that airs on NPR. Local NPR stations often are also very interested in airing essays from listeners, although they aren't likely to pay you anything.