Tails or tail like structures on a cell are usually for movement. Sperm cells have a tail that allow them to move to fertilise an egg cell.
Sperms cells also have a high number of mitochondria to provide the energy needed to swim. Some bacteria cells have a tail or “flagella” that allows them to move through fluids towards food and nutrients.
A fatty layer around a cell is for insulation or protection. Nerve cells have an insulated layer surrounding them allowing them to transmit the electrical signal faster and with greater efficiency meaning that the signal is processed quickly.