The delay between the moment you press the shutter release and the moment that the shutter actually opens is called shutter latency. (It is also sometimes referred to as shutter lag.)
The most prominent cause of shutter latency is the time necessary to move the reflex mirror out of the way in an SLR, before the shutter can be opened. Hence, shutter latency is virtually nonexistent among mechanical rangefinder, TLR, field, and view cameras. With electronic versions of these cameras, however -- and especially on digital cameras -- there is some shutter latency regardless of the presence or absence of a reflex mirror.
Autofocus cameras can often introduce an additional focus delay, although this can sometimes be avoided by initial pressure on the shutter release, which causes some cameras to focus and set the exposure, before full pressure fires the shutter.