I am a cell. I am long and thin. I reach all the way from the brain to the tip of a finger. I have a special coat of fat that helps me do my job. My job is to send electrical signals from one part of?

It sounds like you are a neuron, specifically a motor neuron. Motor neurons are long and thin cells that transmit electrical signals from the brain to muscles to control movement. The special coat of fat you mentioned is likely the myelin sheath, which helps conduct signals more efficiently along the neuron's axon.