How does a neuron send a message to nearby or another neuron?
Well electrical charges are released from the axon terminals where they travel down through the axon which then triggers neurotransmitters that then jump through the synapses onto another cells dendrite's which uses receptors to process the message.
This process is know as action potential. It is created by depolarization. Neurons allow positive and negative ions to enter and exit the cell. Ions such as potassium and sodium level the different charges with in the cell. Normally the inside of the cell is more negative on the inside but since the cell membrane isn't static, the charge changes. Different types of neurons release different types of neurotransmitters. Theses neuromtranmitters will will be released from electrical charges to jump through the synapse which will lead to a post- synoptic neuron.
What is that process called?
You also have to know the threshold which determines the amount of input needed to cause action potential. Once the action potential if fired then the electrical impulse will travel down the axon.