Climate sensitivity is a measure of the equilibrium global surface air temperature change for a particular forcing. It is usually given as a °C change per W/m2 forcing. A standard experiment to determine this value in a climate model is to look at the doubled CO2 climate, and so equivalently, the climate sensitivity is sometimes given as the warming for doubled CO2 (i.e. from 280 ppm to 560 ppm). The forcing from doubled CO2 is around 4 W/m2 and so a sensitivity of 3°C for a doubling, is equivalent to a sensitivity of 0.75 °C/W/m2. The principal idea is that if you know the sum of the forcings, you can estimate what the eventual temperature change will be.
We should underscore that the concepts of radiative forcing and climate sensitivity are simply an empirical shorthand that climatologists find useful for estimating how different changes to the planet’s radiative balance will lead to eventual temperature changes. There are however some subtleties which rarely get mentioned. Firstly, there are a number of ways to define the forcings. The easiest is the ‘instantaneous forcing’ – the change is made and the difference in the net radiation at the tropopause is estimated. But it turns out that other definitions such as the ‘adjusted forcing’ actually give a better estimate of the eventual temperature change. These other forcings progressively allow more ‘fast’ feedbacks to operate (stratospheric temperatures are allowed to adjust for instance), but the calculations get progressively more involved.
Secondly, not all forcings are equal. Because of differences in vertical or horizontal distribution of forcings, some changes can have a more than proportional effect on temperatures. This can be described using a relative ‘efficacy’ factor that depends on the individual forcing. For instance, the effect of soot making snow and sea ice darker has a higher efficacy than an equivalent change in CO2 with the same forcing, mainly because there is a more important ice-albedo feedback in the soot case. The ideal metric of course would be a forcing that can be calculated easily and where every perturbation to the radiative balance had an relative efficacy of 1. Unfortunately, that metric has not yet been found!