What is the relationship between carbon dioxide levels in the atmosphere and global warming?
As carbon dioxide levels in the atmosphere increase global warming increases, assuming no other changes.
Apart from water vapor, carbon dioxide is the most abundant greenhouse gas in the atmosphere. Greenhouse gases allow energy from the sun to pass through and reach the Earth.
This radiation is primarily of wavelengths in the visible light spectrum (about half a micron). When the energy reaches the Earth it heats it up. The heated Earth is actually releasing energy as heat, but it is in a different wavelength (averaged at 11 microns). We refer to energy around that wavelength as infrared. Greenhouse gases block energy at that wavelength.
This is what the balance looks like, it's called the solar budget. It is important to note that the total amount of incoming radiation is equal to the total amount of outgoing radiation. Greenhouse gases (indicated by the 15%) are necessary to maintain the balance.
If you look at the 15% and imagine a great increase in the amount of those gases, eventually that 15% will become 16%. That means that 1% of the incoming radiation is not balance by outgoing radiation. This will result in a heating of the Earth so that the amount of heat released by the Earth will increase so that the amount absorbed by the greenhouse gases becomes 15% again. Now we are back in balance until the numbers change again.
For example, lets say the heat released by the Earth was 100 units, and to balance 15% (15 units) are absorbed by greenhouse gases. If 16 units are absorb then the heat of the Earth goes up. It isn't until the Earth is releasing around 106 units that the 16 units that greenhouse gases are absorbing becomes 15% and we have a balance again. Obviously for the Earth to be releasing 6 more units of heat the overall temperature of the Earth is going to have to go up.
No one disagrees with this at all. The point of debate regarding this is there are a lot of other numbers in that balance and each represents something else that impacts on the budget. Since there are so many other factors some argue that the greenhouse gas factor is somewhat less important.