Final answer:
The part of a neuron that transmits messages to other neurons or to muscles and glands is known as the axon.
Step-by-step explanation:
The neuron fiber that passes messages through its branches to other neurons or to muscles and glands is the axon. An axon is a long, thin extension of the cell body that transmits nerve impulses away from the cell body and toward other cells, such as other neurons, muscles, or glands. The axon may cover great distances in the body and has branches at the end known as axon terminals. It is at these axon terminals where synapses occur, allowing the neuron to transmit signals to other cells. When a neuron is sufficiently stimulated, it generates an action potential that propagates down the axon to its terminals, causing the release of neurotransmitters that then stimulate the next neuron or target cell.