Short answer
Neurons are ‘born’ when a progenitor cell splits into two: one new progenitor cell and one neuron-to-be. This newly formed neuron first travels to its destination in the brain. Then, using chemical and mechanical cues, it develops into a mature neuron with a cell body, an axon, and dendrites.
Longer answer
Neuronen zijn cellen in onze hersenen die gespecialiseerd zijn in het transport van informatie over lange afstand, in de vorm van elektrische signalen. Neurons are cells in our brain specialized in the transport of information over long distance, in the form of electrical signals . Like most other specialized cells in the body, neurons originate from the division of a progenitor cell into two daughter cells: one new progenitor cell and one neuron-in-the-making. While most neurons in our brain were born and grown during our brain development, in certain brain regions, like the olfactory bulb or the hippocampus, neurons are being created throughout life.
After the division of the progenitor cell, the round daughter cell that will eventually become a neuron needs to migrate to its final location on the brain. When it found its place, it needs to develop three basic parts: a cell body, an axon, and dendrites. Neurons receive information from other neurons on their dendrites, analyze the incoming information in the cell body and send out information through their axons to other neurons or muscles. (Read more about this communication here).
The basic parts of a mature neuron. 1 = dendrites, 2 = cell body (nucleus), 3 = axon, 6 = synapse. This answer does not discuss numbers 4 (Schwann cells or myelin) and 5 (Nodes of Ranvier); they enable the very fast communication of neurons. Image source: https://commons.wikimedia.org/wiki/File:Simple_neuron_scheme.svg
As the first step to become a mature neuron, the neuron will start growing long projections from its cell body, also known as neurites. Shortly after, one of the neurites will become an axon and all other projections become dendrites. The axon grows by following chemical and mechanical cues, and in humans, it can reach lengths ranging from a few centimetres to up to a meter. (Read here whether this is different in dinosaurs!). At the end of the axon, it forms synapses onto muscles or the dendrites of other neurons. When an electrical signal passes through the axons, neurotransmitters are released at the synapses. This is a chemical signal that can trigger other neurons or muscle to start another electrical signal. The dendrites are much smaller, only a few hundreds of micrometres long. But do not underestimate them! All dendrites of a single neuron can connect up to thousands of other neurons via synapses. So all these different parts of the mature neuron are crucial to fulfil its important function of communicating information in our brain.