This question has bothered me a little because it lacks a definition of "brain." There are several.
This is the first line from Zb's link:
"I am trying to find out which organism has the smallest brain. By brain I mean neuron, cluster of neurons, or anything resembling a neuron."
Obviously that's what the excerpt she quotes is based on.
Most sources differentiate between the vertebrate and invertebrate brains:
"2. Zoology. (in many invertebrates) a part of the nervous system more or less corresponding to the brain of vertebrates."
(I'm assuming most readers here will have at least a dim idea of what the vertebrate brain is. :-] It's that gray jelly-like mass that doesn't function before 6:00 am unless jump-started with coffee.)
I just can't help but pass along this 4-min. video clip called "How to dissect a fruit fly brain." It doesn't live up to its name, though, ending just as the brain is finally exposed after the fruit fly is literally torn to bits with tweezers under a microscope.