An analog video signal is basically a waveform. It's 100% time based, and one frame takes a specific length of time to transfer, since that is how long the wave is.
The wave itself takes a certain amount of bandwidth, which is basically how much data is held in that wave. It's possible to reduce the amount of bandwidth required through various filtering techniques.
Analog video only really has the concept of "now" - the single pixel that is being displayed at that moment.
Conversely, a digital video signal is an interleaved data stream. One of the sub-streams is the picture stream. This is a frame-based stream, where each frame of the video is treated as an individual entity. It's this concept of the frames that allows video compression. Digital video has the concept of "this frame" rather than "this pixel", so it can compare neighbouring pixels in all 3 dimensions (not only the up/down left/right 2 dimensions of the frame, but also the third "time" dimension, comparing with past, and even future, frames).
An analog video signal can be fairly easily converted into a digital format through the use of a frame grabber. It can then be compressed just like any other digital format.
A good analogy would be audio. Compare an old audio cassette with an MP3. When you're playing a cassette the tape is moving past the read head at a set speed, and the read head converts the magnetism on the tape at that specific moment in time to a movement of the speaker.
Conversely, with an MP3, chunks of data (again, they're called frames) and decodes them into an audio waveform for playing through the speaker.
(note: this is a vastly simplified description, and as a result is completely wrong ;) )