Get an agent. With a polished manuscript you will have the best luck getting your writing published if you have an agent. Send it to different publishers with a cover letter summarizing your work. You have to be able to sell yourself in less than a page, and your manuscript must be clean and ready for review.
Pitch to local and national magazines. Make sure you are succinct and clear about the article you wish to write for them, and you may have to send many pitches to get a writing deal. Nonfiction writing is needed on every topic under the sun, but make sure you know the voice of the magazine you are pitching to. Once you get published once, it will be easier to get published again and again.
Send out any essays you have written to newspapers, magazines or online publications. There are lots of publications that need nonfiction writing and are looking for works to be published. Make sure your piece is polished and ready before you send it in.
Submit your nonfiction writing to contests. Prizes for these usually include money and getting published. Read all of the fine print regarding submission deadlines and submission requirements before sending these out.
Connect with other writers and publishers. Getting nonfiction writing published can be difficult due to the competition out there. Network and find connections in your area to increase your chances of success.