Streaming is a "progressive" download where you watch/listen without actually dowloading the content to your device. Think of it like radio: your player simply receives a signal (or stream) but you never actually save the data to the device.
Of course, via the plane wi-fi, you can connect to your home PC or cloud storage provider (done it personally with Dropbox and LogMeIn for document retrieval). Now, theoretically you could probably connect to your PC at home or a cloud storage provider via a plane wi-fi to start a video stream to your mobile device. Realistically however, due to bandwidth restrictions, you could never really get the data to stream properly. Downloading to your mobile device for offline viewing is really the only option today.
At home on wi-fi, streaming from a PC or NAS to your mobile device (using aforementioned video streaming apps like Air Video, Plex, Video Stream, etc) is very easy. Frankly, DLNA from your PC to TV or DVD is the same concept.