| Contrary to the popular belief of many students in introductory physiological psychology, this is the giant axon of a squid, and not the axon of a giant squid.
That said, the reason the giant squid axon is important is that its study provided many of the initial insights into how ion concentration gradients and changes in membrane permeability produce electrical signals in neurons. The axons of these nerve cells can be as large as 1mm in diameter -- hundreds or even thousands of times larger than mammalian axons. Squid axons are large enough to allow scientists -- or for that matter, students (I've done it) -- to perform experiments that would be impossible on most other nerve cells. For example, it's not very hard to insert wire electrodes from a multimeter inside the giant axon and make measurements. This ability yielded the first intracellular recordings (as early as 1940) of action potentials in nerve cells and the first experimental measurements of ionic currents producing them. The axon's size also makes it practical to extrude its cytoplasm and measure its ionic composition.
Giant neurons probably evolved in squid because they enhanced survival: the neurons participate in a very simple circuit that activates the contraction of the mantle muscle, producing a jet propulsion effect allowing the squid to move away from predators at remarkably fast speed. Larger axon diameter allows faster conduction of action potentials, so squid probably have these huge cells to escape faster. | 
