What is the relationship between CO₂ level and temperature?

Swedish scientist Svante Arrhenius (1859-1927) was the person who quantified the effect of atmospheric CO₂ on temperature. The results were presented in this paper.

The many calculations in the 1896 paper include estimates of the required increase or decrease of CO₂ required to move the climate into a different state.

Apart from his calculations, Arrhenius also addressed important questions such as this one: "Is the mean temperature of the ground in any way influenced by the presence of heat-absorbing gases in the atmosphere?"

He also made important statements about water vapour as a greenhouse gas, such as this one "From observations made during balloon voyages, we know also that the distribution of the aqueous vapour may be very irregular, and different from the ideal mean distribution". This emphasises the point that water vapour differs from CO₂ as a greenhouse gas in that water exists in gas, liquid and solid forms in the atmosphere and is continually cycling in and out of the air. It is therefore distributed in a highly uneven fashion and is uncommon in the upper atmosphere.

Empirical Evidence

A very good summary of the empirical evidence can be found here.

Apart from more details of Arrhenius's calculations and research, it also includes more about the role and properties of water vapour.

But key to the explanation of CO₂ and its warming effect is the discussion of the greenhouse effect. As pointed out in the detail, many other theories/options "fall by the wayside", but the greenhouse effect has been demonstrated and measured and is the one that "fits the numbers".

The Intermediate and Advanced sections at the linked document provide much more detail of both the theory behind the greenhouse effect and the many observational verifications of both its existence and quantification.

The greenhouse effect can be demonstrated and measured, and it explains the warming that the Earth is currently experiencing..