Four Degrees Celsius Would Be Catastrophic

Here's the basic picture,

•    350 ppm CO2 = 1°C warming (a warming we can live with, see
•    400 ppm CO2 = 1.5°C warming (things get worse for an increasing number of people)
•    445 ppm CO2 = 2°C warming (dangerous warming, but the best we can hope for)6

      2°C warming is a guardrail beyond which changes become catastrophic
•    560 ppm CO2 = 3°C warming (really bad: and positive feedbacks accelerate)
•    700 ppm CO2 = 4°C warming (To be avoided at all costs)7
6. (PDF download) Hansen et al. state that 2°C warming could be dangerous. 
7. (PDF download) Turn Down The Heat: Why a 4°C Warmer World Must be Avoided. 

We passed 400 ppm CO2 last year. 
At current emission rates that are increasing CO2 at a rate of 2 ppm/year, we will blow past the budget for staying below 2°C warming within 20 years.  The only possibility for avoiding 2°C warming is for humans to alter their behavior and reduce energy use while we bring low-carbon energy sources online. The Paris Accord is one attempt to motivate governments to move in this direction, but individuals should also move on their own to reduce personal carbon emissions. There simply is no time to waste in waiting for governments to motivate action.
 For those who see a 2°C target as too ambitious, and prefer a longer-term approach of focusing on 3°C as a more attainable target, it is important to note that we may not have an option to stop the warming at 3°C. By that point positive feedbacks may drive the temperature higher, no matter what we do. There are large uncertainties about how the climate will respond as global temperatures increase, but natural feedbacks will certainly increase with increasing warming, making our emissions reductions less effective at controlling global warming as the planet continues to warm. 
That is it exactly.   A 3°C world doesn't only mean a drastically bad world to live in, it also means that we are unlikely to stop there no matter what we do.  2°C will be bad enough, but 3°C presages a much, much worse world.
This is why there is an urgency to meeting a 2°C target. A carbon budget for staying below 2°C is the last carbon budget for which there is a consensus among scientists that we have a reasonable expectation of controlling our own destiny. Beyond that warming, in addition to escalating negative effects, it is unclear how much control we will have over the climate.
And a 4°C means the end of civilization, and very likely the end of the human race.

But don't lose all hope.
There is a growing realization that holding warming to 2°C warming may not be possible (see Glen Peter's excellent summary on this topic). Does this mean it’s too late and we should just give up? If you are driving a car and you suddenly realize you are going to run into a brick wall, no matter what you do, at what point is it “too late” to put on the brakes? At the point you put on the brakes you will avoid something worse. So even if it is too late to stop the warming at 2°C, remember that these are just numbers, there is uncertainty about the effects associated with these numbers, and that we will always avoid something worse later, by taking action now.