There are several ways. One way is right here on HubPages. Research topics and write your articles. Then turn them into HubPages.
Another way is to join places such as ehow and associated content. You can earn money by accepting assignments on content that is posted on each site. You can also earn money off of articles you write that cover your own subjects.
Finally, you can become a freelance article writer. There are many, many Internet Marketers and Web Designers who need content. You can join places such as The Content Authority, eLance and oDesk.
There are others, but these three are the ones that come to mind when you want to make money online by writing articles. Just remember, you're not likely to become rich doing it, but if you stay at it a while you can build up some nice residual income.