How does the earth's atmosphere relate to global warming?

The Earth is in balance with the sun. That means that the amount of energy the Earth receives from the sun is the amount of energy that the Earth radiates into space. This is called Earths energy budget.

If you look you will see that the 100 percent that is started is also what is radiated by adding the numbers of all the arrows that point back out into space.

The way that global warming works is the energy that comes from the sun is primarily in the visible wavelength on the electromagnetic spectrum. We also can call this short wave radiation. Shortwave radiation passes through things like carbon dioxide and methane and water vapor. Those gases are transparent to short wave radiation.

Of the short wave radiation that reaches the Earth some is reflected based on the albedo (reflectiveness) of the surface the rest is absorbed by the Earth since the Earth is opaque to shortwave radiation. This shortwave radiation heats the Earth.

As the Earth warms it releases it's own radiation, not as light but as heat (infrared). Heat is a longer wave radiation than light. The gases I mentioned above are not transparent to long wave radiation so they trap it. That is the greenhouse effect.

The greenhouse effect is a natural part of the energy budget, but if we modify the greenhouse gas amount in the atmosphere then the 16% of the energy that is absorbed by the atmosphere could become 17%, and then for every 100 units of energy we receive from the sun only 99 is radiated.. At that point the Earth has to warm up until the budget gets back in balance.

I will explain the role of the ozone layer just so you can understand what it does. UV light is even shorter wavelength than visible light. Ozone is transparent to visible light but opaque to UV light. UV light causes things like skin cancer. So the ozone layer is very important but it has nothing to do with global warming.