There are several sites were you can watch a movie without needing to download anything: YouTube, MetaCafe, Vimeo and Daily Motion are four examples.
If there's a particular film one want to see, you'll just have to use trial and error to find the right film. Generally speaking, the older a film is, the more likely you are to find it, providing it has been released on VHS or DVD at some stage.
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: