To Digitize means to represent an analogue object, image or sound in a digital format. Essentially, it produces a digital copy or representation of the original. An example is the scanning of analogue sources, such as printed photos or taped photos, into computers for editing.
A digitizer is, by definition, a device which facilitates this reproduction. A great example of this is the glass screen on a touch screen mobile phone, such as an iPhone. The screen transforms the analogue signal of you touching the screen, into a digital signal that the phone can understand and therefore carry out the desired command. This is an example of a LCD Digitizer.
A video digitizer could be a device or software program which converts an analog video input into a digital video output. Used, for example, to save an analog TV image to a hard drive.