What are the three parts of a negative feedback loops and what are there function?

I would say the error, a gain and the integrator.

The error is the difference between the output and the desired output.

The gain is the % of the error that is fed into the integrator.

The integrator is the current state of the plant, ie, that which you wish to control.

1)  Amplifier with gain -A
2) Feedback network with transfer function H
3) Adder at the amplifier input.

The amplifier output is fed through the feedback network to the adder at the amplifier input.  If the input signal is V, and the feedback network ouput is Vf, the amplifier input is V - Vf.  The amplifier output is then  Vo = -A*(V - Vf).  But Vf = Vo * H

Vo = -A*(V - Vo*H)

Vo - Vo*H*A = -A*V

Vo = -V* A/(1 - H*A)

If H is a simpe attenuator of value k, then

Vo = -V*A/(1 - k*A)

However, H can be any complex transfer function including integrator, differentiator, or frequency-selective network.