The Earth is getting warmer because people are putting greenhouse gases into the atmosphere. This is called the greenhouse effect because a greenhouse stops some of the sun's heat from escaping, and these gases stop some heat escaping from Earth. The greenhouse gas which makes the most warming is carbon dioxide (short name CO2), and burning coal is the biggest thing which puts it in the atmosphere. Climate sensitivity means how much the Earth will warm when a certain amount of CO2 is released into the atmosphere. Usually it means how much warmer the Earth would be if the atmosphere had twice as much CO2 as the year 1750, before people started burning lots of coal. Scientists think that when CO2 in the atmosphere doubles an extra 4 joules of heat gets trapped every second on every square metre on Earth, and that after a long time the Earth will be about 3 °C hotter.

How sensitive is the climate?
Climate scientists don't know the climate sensitivity very well but they think it is probably between 1.5 °C and 4.5 °C. So if the atmosphere had twice as much CO2 as it had in 1750 then Earth would likely be between 1.5 °C and 4.5 °C warmer, after thousands of years.

How do we know?
There are three ways to try to find out the climate sensitivity. The first way is to look at the temperature measurements made each year since 1750 and compare them with measurements of how much  greenhouse gas was in the atmosphere at the same time. The second way is to try and work out how much greenhouse gas there was and what the temperature was in the distant past before people started measuring them. For example by measuring how much CO2 is in tiny bubbles of air trapped deep in ice sheets thousands of years ago. The third way is to make models of the climate system in computers.

Why is it important to know better how sensitive the climate is?
The Paris Agreement is to keep global warming below 2 °C. If the climate is very sensitive this will be very difficult, so maybe people and countries should change to using cleaner energy more quickly. But if the climate is not very sensitive perhaps it would be a waste of money to stop using the coal burning power stations which are still almost new. So it is very important to find out more exactly what the climate sensitivity is.

Why is it difficult to find out?

It is difficult to find out how sensitive the climate is, because after the greenhouse effect makes the Earth a little bit warmer there are a lot of different kinds of feedbacks, which can make the warming faster or slower. For example the warmth evaporates more water from the sea, and this water vapor is itself a greenhouse gas, which makes the Earth even warmer. Scientists are fairly sure that most feedback makes the warming faster: but because the feedback is so complicated they don't understand very well, and sometimes their computers are not fast enough to do enough calculations to get a good answer. The project where they work together to understand better is called the Coupled model intercomparison project (CMIP). One of the things they are trying to understand better is clouds, because they might make a big difference.

Different ways to talk about climate sensitivity
People who are thinking about the 21st century might talk about climate sensitivity differently to people who are thinking about thousands of years in the future.

When the CO2 stops increasing
If the CO2 increases gradually by 1% each year the transient climate response (TCR) is the increase in temperature by the year when it has doubled. This is likely between 1 °C and 2.5 °C. Because this is a bit like what might happen this century this might be the most useful way for most people to talk about climate sensitivity when they are talking about the Paris Agreement.

When the temperature of the ocean balances the temperature of the atmosphere
But most of the extra heat warms up the oceans, and after the CO2 has stopped increasing some of this heat leaves the oceans and keeps on slowly warming up the atmosphere for thousands of years. The equilibrium climate sensitivity (ECS) is the temperature the Earth would get to once the atmosphere had got back in balance with the oceans, if the CO2 stopped increasing but stayed at the double level.

References

Other websites 
 What is ‘climate sensitivity’? Met Office
 How scientists estimate ‘climate sensitivity’ Carbon Brief

Climate change