Originally Posted by Brovo
Piracy: Making an illegal copy of something to keep for as long as the user wishes.
Streaming: ^ Not subject to the above.
Here is how it works, at least according to the DMCA.
If the film in question was acquired through pirating and then distributed through streaming, then yes, it's piracy.
If the film in question was acquired through recording it on a TV channel, then it may or may not be piracy according to which country you live in and how they interpret the DMCA.
If the film in question was acquired through completely legal means, I.E. was purchased as a DVD collection for example, and then put up for streaming, then it's treated no differently than if someone brought a couple friends over to watch his copy of a legally acquired film. I.E. It's not piracy.
Depending on the country, where the servers for the streaming site are hosted and where and when the film in question was copywrited, it may or may not qualify as copywrite infringement. However, copywrite infringement is generally not regarded with the same vehement spite as piracy. Some companies even treat streaming sites as free advertisement.
In Japan's case, a streaming site is usually not frowned upon, especially if it brings their product to a foreign audience.
tl;dr: If streaming content that was copywrited was illegal, then Youtube would not exist. Seriously. It would then be illegal to put up Lets Plays, clips of Gears of War, or anything without express permission of the copywrite holder.
The copywrite holder does have the right to call for a takedown if it sees something which it deems as ultimately harmful.
Essentially, if it's not taken down, and the source which uploaded it didn't acquire it illegally, then it's not illegal to view a streamed source of television, movies, or anime.
There you go.
Copywrite 101.