Desktop client for arduino
The easiest way for beginners to do this is to write a program that uses a com port to communicate with the microcontroller. I have never used an arduino, but I know they have the ability to plug in a USB cable to them that then creates a serial port on your computer. This program can be written in any language that can communicate with a serial port (java, c, c++, etc).

So, once you have your com port connected, it is just a matter of writing and reading serial on both ends. You have the freedom to use what ever communication protocol you wish. By this I mean you get to determine what is in the data packets being sent. This will very much depend on what format the camera you are using gives as well as what you want to use the video for.

For example, if you have a raw video stream, you could create a communication protocol that gives a header which includes something like frame number and frame length (number of bytes) and then possibly a footer that includes a checksum for the whole image. Or you could scratch the checksum and header and just mark each row. It's all up to you.