There are several sites were you can watch a movie without needing to download anything: YouTube, MetaCafe, Vimeo and Daily Motion are four examples.
Whether the makers of the film have granted permission for their work to appear on those sites is another matter altogether!
Films For Free on Public Domain
A number of films have entered the public domain, meaning anyone can watch them for nothing at anytime.
These are films where copyright has lapsed and not been renewed or, as was the case with Night of the Living Dead (pictured), the copyright notice was accidentally left off the credits. You can watch such films, entirely legally, at the Internet Archive.
Here's one such public domain film, The Little Shop of Horrors, starring a young Jack Nicholson:
