I have heard that information is sent between the brain and peripheral nerves via electrical pulses or signals, but I don't understand how they create them in the first place.
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.
Generating an action potential
Ok, so how do these parts come together to create an electrical impulse?
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.