The global weather system is complex, and I'm really not going to do it justice in a couple of paragraphs. However, we can get to a simple enough outline that will serve to show us where the problems are. The elevator version, then, is that the sun heats the air, and the ground. The sun's input is concentrated around the equator and, until recently, we have had a stable pattern of hot air rising and flowing north and south to the poles, by which time it has cooled and can flow down to the surface. The earth's rotation, and the different heating and cooling effects of the oceans and continents, makes this circulation more complicated than this sounds and the disturbances from these and other effects are what we see as weather. The global temperature this system works at is affected by our atmosphere through what is known as the greenhouse effect: the light from the sun warms the surface; the surface warms the atmosphere; and the atmosphere warms the surface even further. This works because some of the atmospheric gasses are able to capture radiated heat from the surface. These are greenhouse gasses, and carbon dioxide (CO2) is the most significant of these. The more CO2 there is, the warmer it gets and the balance between heat loss to space and heat capture by CO2 keeps the earth at a habitable temperature. This balance has been reached over the lifetime of the Earth: aerobic plants that use oxygen to give them energy to grow generate CO2 as a natural part of that process, and photosynthesis uses uses CO2 to build structure, returning oxygen to the atmosphere. As plants using these processes have evolved they have contributed to the stabilisation of the atmospheric temperature.
This has been all fine and dandy for a few million years. Recently, though, humans came on the scene and started to use fire. Gradually our ancestors found that fire could do all sorts of interesting things, like making glass, refining metals. On top of this, more recent ancestors discovered capitalism, energy driven production, and competition. The result has been cheap flights, mobile phones, and extra CO2 in the atmosphere. We see this as a sharp, and continuing rise in global average temperature starting around about the industrial revolution1.
If things continue to warm up we can look forward to rising sea levels as the ice caps melt, disruption to the weather patterns, resulting in extremes of heat and cold, parts of the world becoming uninhabitable due to lack of water and high ambient temperatures, extinction of cold climate creatures and plants as the poles warm, potential migration of hot climate creatures and plants to once temperate regions, expansion of unusable desert lands into once temperate regions, and, ultimately the possible destruction of all life on the planet.
Is that too extreme? Maybe, maybe not. For one thing, the warming effect is delayed; any excess CO2 introduced now will take years to show up as a raised temperature. That means that the temperature will continue to rise even if we stop making things worse, but it shouldn't be too much2. On the other hand, any cooling effect is also delayed, so removing CO2 will not have an immediate effect. What is more of a problem is that there are various feedback loops that work against us: ice caps reflect sun light and help to keep us cool; plankton growth in the oceans absorbs CO2, but high CO2 makes the oceans acidic, killing the plankton; and there are others. This means that if the temperature rises above some limit then we have no way back.
All in all, governments have accepted3 that a maximum increase in global average temperature of 1.5 °C is an appropriate target. This is a last ditch figure. The IPCC report2 notes that we are already suffering from adverse effects, species loss, extinctions, loss of land and so on, and these effects will get worse, but the target of 1.5 °C is significantly better than an earlier alternative target of 2 °C.