So, its probably not the most prevalent neuroscience myth out there, but I know a couple people who get really irked when they see the "firing" of neurons portrayed in movies and tv. Some really egregious examples can be found in the openings of the Spiderman movies and in the movie Deep Blue Sea, from which I have a couple of frames below, where the yellow arrow is showing an easily observable (well if my image quality didn't suck) lightning bolt extending between to very distant processes as the neuron "fires".
So what's wrong with this picture? Well neuronal signaling does involve electrical signaling, but it is electrochemical signaling, with the transmission of signals between cells being the chemical portion of the electrochemical. Also, synapses, the spaces between brain cells where the electrochemical signals are relayed from one cell to the next, are really, really small. This depiction makes it seem like the gap between cells is really far when, in actuality, you couldn't fit a human hair through a synapse. (The average diameter of a human hair is around 1/10th of a millimeter. Though the distance across a synapse is likely variable depending on the types of neurons, or the part of the nervous system, or the species in which you are looking, the synaptic space is usually on the order of tens of nanometers, with an average of about 50 nanometers, which is roughly 2000 times thinner than a human hair.) Synapses are so slim, in fact, that you can't even see them at the highest possible magnification under a light microscope.
As for understanding how signals are transmitted across a synapse, we have to talk a little bit about the cells that are doing the communicating. Neurons (or "nerve cells") are specialized cells that are found throughout the brain and the rest of the nervous system. They have a cell body and a nucleus like all of the other cells in your body, but they also have processes that extend out from the cell body like branches from a tree trunk, and these processes allow the cells to communicate (via electrochemical signaling) across great distances (well, great distances for a cell anyway... in humans, some peripheral neurons can extend to be over 1 meter long) Ultimately, all this communication gives rise to sensation, perception, thought, movement, and pretty much all of the main functions we associate with the nervous system. And, for the most part (though this is obviously oversimplifying), the really long branches of the cells (called axons) act like wires carrying electricity. They even have an isulated covering called myelin that wraps around the "wire" like the plastic that covers and insulates the wires running in your house or the power cord to your computer. An electrical signal is propagated in a neuron from the cell body (where the nucleus is housed) down the axon, until it reaches the end of the axon, called the terminal (like a bus terminal, this is the "end of the line"), where it needs to cross the synapse and activate an electrical signal in the next cell in the circuit.
To help visualize the terminal, synapse, and neurotransmitter release, here's an image from brainconnection.com... (which is a great place to go if you want to learn more about the brain and other neuroscience topics).