Short answer- all of it. It's a common myth that only 10 or 20% of the brain is used. From an evolutionary standpoint, having an organ which only used 20% of it's capacity would be incredibly wasteful and would have favored a much smaller brain which used less energy. Short answer We only use 90% at a time.
A better question might be "how much of the brain is used at once?" Just looking at a computer screen, reading the text, and typing a response is going to be using significant portions of the occipital lobe (visual regions), motor areas (typing), temporal regions (language), and pre-frontal areas (planning). If you put someone in an fMRI scanner (used to image brain activity) while they typed away at that task, you might get 30% lighting up as being more active than when they were just lying there doing nothing. But those areas are simply *more* active during a task than when you're doing "nothing". But even when you're doing "nothing" you're still looking at things, thinking about what you'll have for lunch and so on- the brain is always active. True, not every neuron in your brain is firing all of the time, but it's fair to say that the whole brain is being used.