The human brain consists of about one billion neurons. Each neuron
forms about 1,000 connections to other neurons, amounting to more than
a trillion connections. If each neuron could only help store a single
memory, running out of space would be a problem. You might have only a
few gigabytes of storage space, similar to the space in an iPod or a
USB flash drive. Yet neurons combine so that each one helps with many
memories at a time, exponentially increasing the brain’s memory
storage capacity to something closer to around 2.5 petabytes (or a
million gigabytes). For comparison, if your brain worked like a
digital video recorder in a television, 2.5 petabytes would be enough
to hold three million hours of TV shows. You would have to leave the
TV running continuously for more than 300 years to use up all that
storage.
From: http://www.scientificamerican.com/article.cfm?id=what-is-the-memory-capacity
I'm no medical expert by any means, but according to this, roughly 2.5 petabytes. This seemed a little high so i looked some more.
The brain has about 100 billion nerve cells, so at least that many
bits (about 10 gigabytes) could be stored, assuming the brain uses
binary logic. But it probably doesn't do so. Instead, information is
believed to be stored in the many connections that form between the
cells. This is a much larger number: Current estimates of brain
capacity range from 1 to 1000 terabytes! It would take 1,000 to 10,000
typical disk drives to store that much information.
From: http://www.moah.org/exhibits/archives/brains/technology.html
This second article explains that if the brain used binary storage then it would be 1-10 terabytes, most likely around 3. But since the brain does not and it stores information in the connections it is much likely a lot more. But this article was written in 2000 I think.
The first article is on the high side, while the second article is on the more realistic side. But the truth is that no one knows because with the technology that we have and understanding of the brain it is impossible to accurately calculate this. from other sources that I have read it looks like between 3-100 terabytes is what is generally accepted. This is a large range, but because we don't know exactly. And a lot of the calculations assume things that we, as of right now, cannot be 100% certain on.
http://www.slate.com/articles/health_and_science/explainer/2012/04/north_korea_s_2_mb_of_knowledge_taunt_how_many_megabytes_does_the_human_brain_hold_.html was another good article.