How powerful a computer is required to simulate the human brain?

H+ magazine wrote an estimate in 2009 that seems broadly comparable to other things I've seen; they think the human brain is approximately 37 petaflops. A supercomputer larger than that 37 petaflop estimate exists today.

But emulation is hard. See this SO question about hardware emulation or this article on emulating the SNES, in which they require 140 times the processing power of the SNES chip to get it right. This 2013 article claims that a second of human brain activity took 40 minutes to emulate on a 10 petaflop computer (a 2400-times slowdown, not the 4-times slowdown one might naively expect).

And all this assumes that neurons are relatively simple objects! It could be that the amount of math we have to do to model a single neuron is actually much more than the flops estimate above. Or it could be the case that dramatic simplifications can be made, and if we knew what the brain was actually trying to accomplish we could do it much more cleanly and simply. (One advantage that ANNs have, for example, is that they are doing computations with much more precision than we expect biological neurons to have. But this means emulation is harder, not easier, while replacement is easier.)

Not just how much, but what kind of processing power : there're specially-crafted dedicated chips, and it has a practical applications, so it's not a lab-only project

The human brain contains about 100 billion neurons ($10^11$) and about a hundred trillion synapses ($10^14$). Each neuron can fire about 100 times a second. If we model the brain as a simple neural network, then it would be equivalent to a machine that requires 1016 calculations per second and 1013 bits of memory.

From Wikipedia

Kurzweil introduces the idea of "uploading" a specific brain with every mental process intact, to be instantiated on a "suitably powerful computational substrate". He writes that general modeling requires 1016 calculations per second and 1013 bits of memory, but then explains uploading requires additional detail, perhaps as many as 1019 cps and 1018 bits. Kurzweil says the technology to do this will be available by 2040.

According to this site here:

Using the NEST software framework, the team led by Markus Diesmann and Abigail Morrison succeeded in creating an artificial neural network of 1.73 billion nerve cells connected by 10.4 trillion synapses. While impressive, this is only a fraction of the neurons every human brain contains. Scientists believe we all carry 80-100 billion nerve cells

It took 40 minutes with the combined muscle of 82,944 processors in K computer to get just 1 second of biological brain processing time. While running, the simulation ate up about 1PB of system memory as each synapse was modeled individually.

Computing power will continue to ramp up while transistors scale down, which could make true neural simulations possible in real-time with supercomputers.

SpiNNaker is a manycore computer architecture designed to simulate the human brain. It is planned to use 1 million ARM processors (currently .5 million). The completed design will hold 100,000 cores

In this video, they showed a completed rack with 100,000 cores emulating 25 million neurons (at $frac14$ the efficiency—it will eventually run 1,000 neurons per core).