As in, what physically travels from our fingertips to our brains?
I’m wondering because I don’t understand how pinching a nerve would affect an electric impulse, which is the only thing I would assume could travel as fast as our nerves do.
I used to design equipment for a neurobiology researcher. I’m no expert, but a little of what he was doing rubbed off. Any mistakes in the following are due more to my bad memory than his knowledge.
Anyway, the nervous system is incredibly complicated. At the most basic level, you have different types of nerve cells. Some respond to pain, some to heat, some to pressure. When they respond they send an electrical signal to the next nerve cell over, and it sends a signal to the one after that, etc. until the signal gets all the way up to your brain. Its horribly slow compared to how we think of electricity on wires, but it works very well.
The complicated part is that when you get to the details, the nerve cells don’t just fire in response to stimulus. At first they are kinda reluctant to fire, then once they really get going they don’t want to stop, even after the stimulus is removed. This causes some weird things that basically result in your nervous system performing integral calculus on the messages coming in. For example, if the stimulus you receive is hot nothing hot nothing hot nothing, what ends up getting to your brain is hot nothing hotter cooler much hotter and not quite cool (it integrates so that the overall sensation feels like its rising).
Your nerves also get a little confused when similar stimulus ends up in the same nerve bundle. This is why your ears itch when you have a sore throat. It’s not confusion in the brain, it actually gets a bit scrambled down in the nerves themselves.
ecg said a mouthful when he says, “the nervous system is incredibly complicated.”
Mildly put, your explanation is good too. I’ll see if I can help at all.
You said an electric impulse and basically that is correct.
It’s a bit more complex though…actually nothing PHYSICAL travels through the nerve cells. They contain chemicals (neurotransmitters) which respond to certain stimuli. What happens when these neurotransmitters become active they build up pressure due to their activity within the cell. This pressure is released to the neighboring cell across a gap (the synapse) which is the transfer of energy (heat) from one cell to the other in an electrical impulse, across the gap it fires the neighboring cell into an excited state which passes the spark to the next cell and so on etc to the brain.
Of course this all happens very fast!
This is about as simple as I can make it and it doesn’t do the nervous system justice as far as an explanation goes but maybe it’ll give some idea of the basic concept. 
Google “action potential”.
As I understand it, although nerve signals have electrical properties, nerves are not simply electrical conductors; the signal travels along the nerve more like a ‘Mexican Wave’ - the neuron consists of a series of electrochemical devices that are triggered in series, each one triggerring the next.
A single nerve cell transmitting a signal works a little like this (very simplified version):
A stimulus causes Sodium (Na[sup]+[/sup]) channels in the nerve’s membrane to open, allowing Na[sup]+[/sup] to rush into the cell (at rest, there is less Na[sup]+[/sup] inside the cell than outside). This has the effect of making the inside of the cell more positive (because of the balance of Potassium (K[sup]+[/sup]) in addition to Sodium).
Na[sup]+[/sup] rushing into one part of the cell will encourage the opening of Na[sup]+[/sup] further down the cell (neurons only transmit impulses one way, in the direction Na[sup]+[/sup] channels will continue to open).
As the wave passes, K[sup]+[/sup] channels open and the K[sup]+[/sup] rushes out of the neuron.
Energy (ATP) is used in the Na[sup]+[/sup]/K[sup]+[/sup] pump to bring the neuron back to its resting state.
So you could say that a wave of depolarization travels down a neuron, or a wave of positive internal charge, rather than electrons or an electrical current.
AFAIK, the electrical impluse cannot travel between neurons. Neurons release neurotransmitters to bridge the synapses (gaps between nerve cells), Dopamine, Serotonin, Acetylcholine, Endorphins, etc.
A somewhat more thorough explanation from the University of Washington.
And a Wikipedia article.