What is the name of the long, thin extension of the neuron that transmits signals to other neurons or muscles?

The axon is a long, slender extension of a neuron that transmits electrical signals away from the cell body to other neurons, muscles, or glands. Axons can range in length from a few micrometres to over a meter, and their diameter can vary depending on the type of neuron and the location within the nervous system. 

Hence, the correct option is the axon.