What is the inheritance of acquired characteristics ?

People (or any other living) do not inherit acquired characteristics.

If your father loves to get involved in body building, you are not going to inherit his large muscles. He worked to enlarge them through exercise but you will have to as well.

This is an old theory by others before we understood Darwin's Theory of Evolution. Others were trying to explain how some animal or plant looks and acts the way that it does.

Plants that live in the desert, have adapted to their environment and the ones that did it best lived longer and had more offspring like themselves.

We say that the environment pressured them or selected the ones that did the best in that environment. If the environment changes, others with other traits may do better.

People with darker skin come from or live in an environment where dark skin is protective against UV light from the sun.
People with lighter skin are not protected and can get skin cancer.

It is true that they can use sun screen but long ago, there was none. They lived a shorter life span and produced fewer offspring.

The opposite is true in places where sunlight is not as strong and lighter skin will be more common. In this case the Vitamin D from the sun is important. This vitamin is necessary to absorb calcium for bone development. Lighter skin does a better job.