The neuron, a type of nerve cell, is the functional unit of the nervous system. The distinguishing feature of neurons, as compared to any other cell type, is the fact that it has protein receptors that mediate the flux of positive and negative ions across the membrane of the cell, which then allows the cell to convert chemical signals to electrical signals, which are eventually "carried" to the brain. Any information that we receive from our external and internal environments, can be coded in electrical signals, called action potentials, and can travel from neuron to neuron until it reaches the appropriate brain center. The only way to encode different information is by varying the frequency of the action potentials, and further, by sending it to different brain regions. The ability to distinguish stimuli depends on where the impulses go, and what part of the brain receives them, not what triggers them.
So let's take a look at one neuron...
Typically (though not always) a neuron has four main regions: dendrites, cell body, axon, and synaptic terminal. See below for animation of what typically happens within and between neurons as nerve cells communicate with each other.
As you can see, there are three major components to neuronal communication: receipt of information, transport of that information down the axon, and delivery of that information to the next axon.
When neurotransmitters (chemical messengers) are released from the synaptic terminal of neuron 1, the neurotransmitters bind to their appropriate receptors on the dendrites of neuron 2. When these receptors are bound by neurotransmitters, the receptors open, and allow ions (positive or negative depending on whether the stimulus is exciting or inhibiting) to flow into the cell.
The cell "adds up" the total influx of charge that it receives from all of its receptors, and notices a corresponding increase or decrease in the membrane voltage. If enough positive charge accumulates (where the voltage increases above a certain threshold point) an action potential is evoked at the cell body. Let's follow this action potential as it travels through the neuron.
First of all, what is an action potential? It is the process where a net positive charge enters the cell, and the cell becomes depolarized (increases in voltage). Why is this important? This increase in voltage will allow other receptors along the cell to open up, because these receptors are said to be "voltage-gated". Meaning, when the voltage increases and reaches a certain point, these receptors will open up and allow specific ions to flow through. Remember that the receptors we talked about before (at the dendrites of neuron 2) were chemically gated, not voltage gated. So when an action potential occurs, it means that part of the axon is infused with lots of positive charge. This positive charge diffuses to nearby regions of the axon, where the voltage-gated receptors there (now being sufficiently depolarized with all this positive charge) will open up and illicit a new action potential. The influx of even more positive charge will again spread to other neighboring areas, and open other neighboring receptors, etc. etc, and in this way the action potential is said to propagate down the entire axon until it reaches the synaptic terminal. Meanwhile, keep in mind that the previously activated axonal regions (that already had an action potential), will neutralize their excess residual positive charge by sending some back out the membrane, and in that way ready themselves for another stimulus.
So as this action potential reaches the synaptic terminal, it activates a set of calcium channels. These receptors are also voltage gated, in the sense that, as the action potential arrives, it brings with enough positive charge to depolarize the synaptic terminal and open the receptors with calcium channels. Calcium (via a complicated and not fully understood mechanism) causes synaptic vesicles (containing neurotransmitters) to fuse with the membrane of neuron 2. After fusing, they release neurotransmitter molecules into the synaptic cleft (the area between adjacent neurons). These molecules will then find their appropriate receptor on neuron 3 and bind with them. This was the same as what happened at the junction between neuron 1 and 2. The process repeats itself all over again until the train of action potential reach the cortical regions of the brain.
Remember that this is a simple picture of events. We followed only the travel of one action potential, but many could be preceding or following that particular one, in varying frequencies. More than one type of neurotransmitter and neurotransmitter receptors exist at each synapse. Furthermore, while we encountered voltage gated and ligand (chemically) gated receptors, there is more to the story. Some receptors allow flow of ions, while others initiate a series of biochemical pathways within the cell, which can really amplify the incoming signal or serve to modify the existing situation in some way. The sheer variety of receptor types and functions is a chapter in its own. Finally, complicated mechanisms exist that can constantly change the nature of a synapse (increase number of receptors, type of receptors, etc), making the synapse a dynamic entity, and a plausible site for information storage and memory.