Check a show's network website. For example, if you want to watch episodes of "The Office", head to NBC's website. Network sites typically have a tab to click on for full episodes. Not all shows are available on any network and you might have to wait a week or so for the newest episode to be available online.
Look at websites devoted to offering TV shows and movies. For example, Hulu offers many old and new movies and TV shows for free. It also offers "Hulu Plus" for a fee, which provides broader content.
Try Netflix. As of publication, Netflix charges $7.99 for unlimited streaming videos. Many recent popular TV shows are available, but the most recent episodes might be available only by adding DVDs by mail, at an extra cost. For example, the first two seasons of the FX drama "Sons of Anarchy" are available online, but the third is by DVD only, as of publication.