The information between the brain and peripheral nerves is sent via electrical pulses or signals, How then does a non-metallic human cell manage to conduct an electrical signal?
This is quite a big question! I'll try to outline the basic view.
First, let's review how neurons signal between each other. The canonical way for a neuron to send a signal to a downstream neuron is by generating an action potential, the "electrical impulse" you have heard of. This action potential causes the release of neurotransmitter at a point where the two cells are very close to each other called a synapse. The downstream postsynaptic cell receives the neurotransmitter signal and converts it into a small electrical signal. If enough of these small electrical signals happen in a short time, they sum together and are likely to initiate an action potential in the second cell and the cycle repeats all along the circuit.
How is the electrical signal generated? The basics of how this works was worked out most famously by Hodgkin and Huxley in 1952. The short story is that the plasma membrane is selectively permeable to ions. Let's build the concept from the ground up.
Imagine a sphere of plasma membrane that represents a simple neuron. For starters, we assume that this membrane is bare lipid with no membrane-associated proteins. Because of the hydrophobicity of the bilayer, charged particles cannot diffuse through the membrane.
The cell is bathed, inside and outside, in a solution containing many ions (charged atoms), including sodium (Na+), potassium (K+), chloride (Cl-), and calcium (Ca2+). As we noted above, these ions cannot go through the membrane without "help".
Now we add an ion pump protein into the membrane which will pump sodium ions out and potassium ions in. This particular pump, the Na-K ATPase, creates an excess of sodium ions outside the cell and an excess of potassium ions inside.
Now we add a potassium ion channel to the membrane. This protein creates a pore in the membrane that only allows potassium ions through. This particular protein's pore is always open. Now things start getting exciting...
What do the potassium ions do now that they can go through the membrane? Ions will move based on the forces created by their electrochemical gradients. The pump created a chemical gradient by putting excess K+ inside, so the K+ ions start to flow out through the ion channels. But K+ ions are positively charged, so when they flow out, positive charge starts building up outside and negative charge builds up inside. This electrical gradient opposes the chemical gradient, tending to pull the K+ ions into the cell while the chemical gradients pulls K+ ions out. The influx and efflux reach an equilibrium at the Nernst potential, where the electrical and chemical forces equal out. For physiological concentrations of K+ ions, the K+ equilibrium potential is about -80mV or -90mV. This means that K+ ions will flow until the outside of the cell is 80-90mV more positive than the inside of the cell. We started at 0mV, so K+ ions mostly flow out.
We now have a membrane potential, a difference in electrical potential between the inside and the outside of the cell at about -80mV (usually closer to -70mV or -60mV in "real life"). In particular, this membrane potential is the resting potential that exists when the cell is not active. We can simplify for now and think of the resting potential as being set by a resting permeability of the membrane to potassium ions, but not to sodium ions. We call this membrane polarized, and thus depolarization is when the membrane potential becomes more positive, and hyperpolarization is when the membrane potential becomes more negative.
Now, we add to the membrane a voltage-gated sodium channel, an ion channel that passes only sodium ions but is usually closed. The voltage-gating means that this ion channel is sensitive to the membrane potential. At the resting potential, the pore is closed and the membrane is still impermeable to sodium ions. When the membrane potential becomes slightly more positive, the channels opens and sodium ions can flow. This channel is also inactivating, so that when it opens it only opens for a short period of time, letting in a limited amount of sodium.
What way will sodium flow when we open this channel? Because of the negative resting potential (-70mV) and the excess of sodium ions outside due to the pump, both the electrical and chemical gradient will drive sodium ions into the cell. The sodium equilibrium potential is usually around +60mV.
To complete the machinery for generating an action potential, we also add a voltage-gated potassium channel to the membrane. It works just like the voltage-gated sodium channel that is also closed at rest and opens when the membrane potential becomes more positive. This channel opens a bit more slowly than the sodium channel does, but it does not inactivate.
Generating an action potential
Ok, so how do these parts come together to create an electrical impulse?
The cell sits at its resting membrane potential, with all of its voltage-gated channels closed. It receives a signal from an upstream cell that causes a slight depolarization. The action potential will initiate when the membrane potential hits the threshold potential.
At the threshold potential, the voltage-gated sodium channels open letting sodium ions flow into the cell. The sodium flux pulls the membrane from the resting potential (-70mV) towards the sodium equilibrium potential (+60mV). These values are far apart, so the driving force is large and the membrane depolarizes rapidly. This is the action potential upstroke.
The depolarization also activates the (slightly slower) voltage-gated potassium channels. The potassium ions flow out and drive the depolarized membrane (about +20mV at the action potential peak) back towards the potassium equilibrium potential (-80mV). At the same time, the sodium channels are inactivating so that sodium is no longer depolarizing the membrane. The repolarization rate is usually slower than the depolarization rate. This is the action potential downstroke.
The whole process of the action potential depolarization/repolarization cycle takes about 2-3 milliseconds in an "average" neuron. Once the cell reaches resting potentials again, the membrane is basically reset. The voltage-gated channels are turned off. The ion pump moves the potassium ions that flowed out and the sodium ions that flowed in. That patch of membrane is ready to fire another action potential!
As a final note, I'll mention that the voltage-gated sodium channel provides a mechanism for the action potential to propagate down the axon. The action potential is initiated in one location of the cell, and creates a depolarization. This depolarization causes the voltage-gated sodium channels in neighbouring regions of the membrane to open and generate an action potential cycle of their own. This is how an action potential travel down axons (and sometimes dendrites too).
So, let us introduce some keywords.
The "electrical pulse" that "is sent from between brain and nerves" is called an Action Potential (AP). This is then propagated along a nerve fiber until the target organ.
Basically, a neuronal cell has a body and several long extended structures that "sprout" from the cell body. Dendrites receive signals from other cells and they convey signals towards the cell body by creating small electrical currents. The axon is a single "sprout" that is usually much thinner and longer than the dendrites and it conveys action potentials from the near the cell body to target cells and organs. Some axons can be as long as 80-90 cm (imagine!)! At the place where axon leaves the nerve cell body there is a small protrusion called the axon hillock.
The AP originates at a special part of the axon called the axon initial segment (AIS). The initial segment is the first part of the axon as it leaves the cell body and sits immediately after the axon hillock.
The electrical pulse is the short electrical discharge, that can be seen as a sudden movement of many charged particles from one place to another. In our cells we have ions of Na+ (sodium), K+ (potassium) and Cl- (chloride) (and in some cases also Ca2+) that constitute these charged particles.
There are two types of driving forces for these particles: besides the potential gradient, e.g. the difference in the total charge in two different places there is also another force called concentration gradient, e.g. the difference in concentration at two different places. These force can point into opposite directions, and thus by exploiting one force (let's say concentration gradient) we can influence another one.
What we need here again is a so-called semi-permeable membrane, this is just a barrier for ions, but only for specific ones. We need this because our main ions -- Na+ and K+ -- are both positively charged. Therefore the cell membrane acts as a semi-permeable membrane, letting K+ into the cells and Ca2+ ions outwards but not the opposite. Therefore we have two concentration gradients: Na+ (outside is the peak) and K+ (inside is the peak).
In order to start the pulse we need to initiate a massive ionic drift from one place to another. This is done by the cell, and the first event here is the drastic change (increase) of the permeability for Na+ ions. Na+ ions massively enter the cell and their charges, moved into the cell, form the upstroke of the action potential.
The protective mechanism of the cell immediately start working against the Na+ invasion and open the reserve shunts -- the K+ channels. K+ leaves the cell, taking away some charge and this is revealed as the decay of the action potential. But potassium channels are generally slower, that is why the decay of the pulse is more steady, not as sharp as the upstroke.
You might be wondering now: what triggers the rapid change of membrane permeability then? There are several factors here that may contribute into this process.
Potential change of the membrane. Sodium and potassium channels are voltage-sensitive, meaning if you manage to change the resting potential of the membrane, formed due to concentration gradients and normally being about -90..-80 mV (millivolts) up to about -40 mV it will trigger the sodium channels. This is how the impulse propagates -- having originated at one place it just decreases the resting potential of the adjacent membrane area, sodium enters the cell there and the AP travels along the nerve. The AIS is the site of AP initiation because this part of the cell has a very high density of voltage-gated sodium channels.
Chemical agents, called neurotransmitters, can be detected by receptors on the cell membrane. Some of these receptors are ion channels themselves and open directly when neurotransmitter is bound. Other receptors act through intracellular signals to open ion channels. This is how the signal appears at the sites of nerve cell contacts -- neurotransmitters, like acetylcholine or adrenaline, just act here as triggers for membrane permeability.