Emotional intelligence is the thought that our intelligence is not just our IQ but also how we navigate our own and others feelings. Someone with a high level of emotional intelligence will be able to recognise their anger and rather than react to it can put it to the side to process what is happening. In comparison a low level of emotional intelligence in the same situation would feel the anger and react to it. This often leads to lashing out.

EQ and IQ have been simplified down to book vs street smart. However this does EQ an injustice. EQ focuses largely on intangible qualities. It is harder to judge because it can not be measured in school as IQ can be. There are no necessarily right or wrong answers to EQ rather it is observed in practice.
For example I had a teacher at school, he was a very, very smart mathematician but could not control his classroom. In comparison with the physics teacher who was also particularly intelligent, but also had a great understanding of how teenage girls acted. He was able to channel that chaotic energy in a classroom towards work rather than working against it.

EQ is important because it effects how we interact with people. It dictates whether or not we respond to an angry message or if allow everyone the opportunity to cool off. This is becoming the dividing factor in employment now, long gone are the days of arguments on the office then down to the pub to have a beer and sort it out. To be able to operate in the modern world one has to be able to navigate their own emotions and make room for others.