Using a website to make money, even if only to support itself, is not an uncommon desire. That's the original question:
Making money on your website is actually pretty easy.
However making lots of money requires work.
There are many, many programs for website advertising. One of the easiest and now one of the most common is Google's Adsense. Adsense, and its companion program Adwords (for advertisers), are the plain-text ads you see on many websites. The giveaway? The phrase "Ads by Goooooogle" above or below the block of ads.
Adsense is easy to sign up for, and once approved will display ads targeted to the page that the ads are displayed on. Adsense is a PPC or "Pay Per Click" system - meaning that the site displaying ads gets paid when a visitor clicks on one of the ads. The actual amount paid varies based on many factors.
There are other approaches including affiliate programs allowing you to advertise and sell items on your site, making a percentage of each sale. Amazon and eBay both have programs as do many other retailers - the keywords to look for in investigating these sites is an "affiliate" or "associate" program.
One caution: advertising and affiliate programs are all well and good, but there's really a bigger issue that prevents most websites from really making a lot of money: traffic. All the advertising in the world is pointless if no one visits your site and if no one sees the ads or buys the products. Traffic generation is well beyond the scope of this article, but there are lots of resources on the internet that can help.