Emotional intelligence is the capability of understanding the emotional needs of both yourself and of others, and guiding yourself and others accordingly. It involves qualities like self-awareness, self-control, motivation, empathy and social skills.
It is different from IQ, which is focused more on intelligence about areas outside skills with people, such as academic (eg. being well educated at subjects like mathematics and science), career-wise (being trained in your profession), and practical/technical (eg. knowing how to build things, being computer savvy). Having this kind of intelligence can make one competent and capable in their career and personal life - but it is not enough when it comes to overall fulfilment in your life, which is where emotional intelligence comes in.
Emotional intelligence is important in order to lead an ideal lifestyle that fulfils all your needs, which requires awareness of these needs and knowing how to manage them, likewise with the needs of others.
Relationships and social interaction, whether with friends, family, partners, work acquaintances and others, are crucial for having happiness and success in your life, and is where emotional intelligence is needed most obviously. Relationships that work out well involves a sense of mutual understanding and a willingness to work towards the most mutually ideal scenarios together. An example of this could be the relationship between a boss and his employees. A boss could have a perfectly organised plan for a company project, but unless he motivates the employees sufficiently, the project may not turn out as amazing as he would have hoped. The boss could use emotional intelligence qualities such as empathy and explain to each of his employees how they will individually benefit from working on the project.
Emotional intelligence is also required for managing your needs mentally and physically. During times when we feel negative emotions, emotional intelligence allows us to recognise these thoughts and act upon them for a positive outcome rather than be consumed by them. For example, someone might feel very angry after receiving poor exam results at school. An emotionally intelligent person would become aware of what they are feeling and start taking accountability by assessing what they can do to improve the situation rather than focus completely on the anger, in this case if the person felt they deserved a higher grade they could dispute with the markers whether they deserved the poor grade or not. Even when the situation is not resolved in an ideal manner, the emotionally intelligent person would know when there is no more they can do, move on, and instead focus on what they could do better next time; for example the person has come to a conclusion that they really were in the wrong and would improve the situation by working harder or choosing to do something else more worth their time.