Short answers, pulled from the story.
Climate sensitivity measures how much Earth's surface temperature rises in response to a doubling of atmospheric CO2. It matters because its value directly determines whether international targets such as the Paris Agreement goal of limiting warming to below 2 degrees Celsius can be met. One study found that halving the uncertainty in the transient climate response could save trillions of dollars.
Transient climate response (TCR) is the temperature change averaged over a 20-year window centered on the moment CO2 doubles, estimated by the IPCC to likely fall between 1 and 2.5 degrees Celsius. Equilibrium climate sensitivity (ECS) is the long-term temperature rise after the planet fully adjusts, including ocean equilibration that can take centuries or millennia; the IPCC Sixth Assessment Report placed ECS between 2.5 and 4 degrees Celsius with a best estimate of 3 degrees.
Svante Arrhenius in the 19th century was the first to quantify warming from doubled CO2, initially estimating 5 to 6 degrees Celsius before revising the figure to 4 degrees. The first calculation using detailed absorption spectra and a computer was performed by Syukuro Manabe and Richard Wetherald in 1967, producing an estimate of 2.3 degrees Celsius, which they rounded to 2 degrees in their abstract.
The committee chaired by Jule Charney and convened by the United States National Academy of Sciences estimated equilibrium climate sensitivity at 3 degrees Celsius, plus or minus 1.5 degrees. According to Manabe, speaking in 2004, Charney derived the 1.5 to 4.5 degree range by taking Manabe's estimate of 2 degrees and Hansen's estimate of 4 degrees and adding a 0.5 degree margin at each end.
The main source of variation is the treatment of cloud feedbacks. Across 27 models contributing to the 2021 IPCC Sixth Assessment Report, ECS estimates ranged from 1.8 to 5.6 degrees Celsius; improved modelling of low clouds drove many estimates higher, while models with the highest values failed to reproduce observed historical warming and were given reduced weight.
Paleoclimatologists use reconstructed temperatures and CO2 levels from past geological periods. Key reference points include the Last Glacial Maximum about 21,000 years ago, the Paleocene-Eocene Thermal Maximum about 55.5 million years ago when temperatures rose approximately 6 degrees Celsius over roughly 20,000 years, and data spanning the most recent 420 million years. The oldest continuous ice core, however, is less than one million years old, which limits direct CO2 measurements for earlier periods.