Depending on what you are writing, the process for how to get articles published can be anywhere from very easy to very challenging. The trick, as with most things, is being savvy and having realistic expectations. But if you choose the right market and do your research, there's nearly always a way to get any article published.
Identify a target market. If it is an article about how to fix toddler sleep problems, find a parenting magazine or website. If it is an opinion piece about an upcoming election, perhaps a newspaper or news site would be the best venue.
Read several recent issues of a publication to get a feel for what kind of articles it publishes and what it has covered recently. If the publication is a website, spend some time browsing the recent articles for the same reason.
Find out the name of the editor at the publication who is in charge of freelance articles. The best person to target in a large publication is usually the section editor whose area is most relevant to the subject of your article. In a publication with a small staff, try the managing editor. Most magazines and websites contain contact information for the staff, although you can also try a directory like "Writer's Market" (see Resources below).
Check to see if the publication has writers' guidelines available for freelance writers interested in submitting articles. If so, follow the instructions in the writers' guidelines to submit your article for consideration.
Send a short note to the editor asking if she would be interested in publishing your article. This is called a query. If you receive a reply indicating interest in your article, send it in as requested. The editor will then either reject the article or make you an offer to publish it.
Consider publishing your article on the Internet without targeting a specific publication. You can try one of the many content providers that offer writers a flat fee for articles or a percentage of ad revenues from the article. You can also start your own blog or website and post your articles there.
Although many writers' resources recommend sending query letters by postal mail, the best method is nowadays is probably to email. You are more likely to receive a reply to a query if it is as simple as clicking the "reply" button rather than if the editor has to go to the trouble of typing and printing a letter. Otherwise, unless the editor has extreme interest in your article, your query letter may simply end up in the recycle bin. Many established publications do not use content from freelance writers at all. Be sure that the publication you wish to target accepts freelance submissions. A simple way to find out is to call the main number for the publisher and ask whether they accept freelance articles.