Climate scientists make predictions about future warming with computer models. The models use mathematical equations to represent our physical understanding of the atmosphere and ocean, the role of life, and possible future scenarios for greenhouse gas emissions. These same models can be used to simulate climates in the past when CO2 levels were higher, and checked against the fossil record (link to page on past climate).
Matt Huber talks about climate models and data from past warm periods:
What past climates say about CO2 and warming.
Climate sensitivity is defined as the expected global warming for a doubling of the level of atmospheric CO2. Modelling based on physics indicates the value lies somewhere between 1 and 5°C, with a middle estimate around 3°C after centuries, when energy balance is reached. If the level of CO2 in the atmosphere suddenly doubled, the initial warming over 10 years would be less than a degree, in part because the build-up takes time and also because the oceans slowly take up the heat, and act as a buffer. However modelling also indicates that if we continue with our emissions today, by the end of the 21st century, CO2 levels could be between three and four times the pre-industrial values, and global average temperature as much as 6°C higher.