We say neurons send electrical messages to each other. Really, they send electrochemical signals. What this means is that chemicals in the body of the cell cause an electrical signal. When electrically charged, chemicals are known as ions. In this context, when thinking about neurons, important ions in the nervous system are sodium, potassium (positively charged), calcium (positively charged) and chloride (negatively charged).
When a neuron is at rest, it is not sending a signal. Even though it is not sending a signal, it is negatively charged in relation to the outside of the cell. This charge (around -70mV) is known as the resting membrane potential. The reason it is negative is that the cell wall only allows some of the ions to leave the cell.
So we know what happens when a cell is at rest, but what happens when it has something to say? The electrical current produced is called the action potential and happens when a cell sends information along its axon away from its soma (cell body). The action potential is caused by a depolarising current, this sounds fancy but all it really means is that something causes the resting potential to move from -70mV towards 0. When the charge reaches around -50mV the neuron fires its own electrical signal. This is known as the threshold level, as if the charge does not reach this the neuron will not fire. All neurons fire the exact same charge, meaning they work very much like a binary system. Like a computer, they have two settings on or off (1 or 0).
The triggering of an action potential is caused when different ions enter the neuron membrane, often these are sent by another neuron in the chain having just fired a charge and released neurotransmitters (chemicals that bond with the axons and dendrites of the cells to stimulate sodium channels to open). Because there are more sodium ions outside of the cell than inside, when the channels open the sodium ions rush in causing the cell to become more positive and depolarise. The potassium channels open to counteract this (remember, potassium is negatively charged) but take far longer to open meaning re polarisation (moving the charge back to -70mV) only happens after the neuron has sent its electrical signal.