The process of conduction across cells happens through the movement of ions, which are small atoms that contain an electrical charge. While there are a few different types of cells that exhibit this flow of electrical conduction, the most common type is the nerve cell, which is often referred to as a neuron.

Our body releases chemicals called neurotransmitters. When these neurotransmitters reach the ends of neurons, called the synapse, they cause a change in the nerve cell membrane. This then causes the molecule sodium to flow inside the cell membrane, which starts the conduction process.

The conduction in a neuron cell is electrical and it is caused by chemicals that are inside the cells. The whole process of conduction in a neuron is the “action potential”, “spike”, or “impulse”. The action potential is an explosion of electrical activity that is created by a depolarizing current, and it occurs when a neuron sends information down an axon, away from the cell body.

The action potential is based on the work of chemicals in the body which are "electrically-charged”. These electrically-charged chemical are called “ions” and in a nerve cells they are potassium (K+) and sodium (Na+). Depending on the state of the nerve cell, which can be sending or not a signal, the ions will cross the membrane of this cell producing electrical forces or voltage. It is the difference of voltage which produces a current and the conduction process.

I found for you a good website where you can see how this conduction process works inside the nerve cells. I also found nice videos where you can watch the process: