Emotional Intelligence is basically the ability to be aware of what we, as a person are feeling and the ability to differentiate between our various emotions. It also includes the ability for a person to know what another person is feeling. For example, in the words of Hermione Granger (HP&OOTP),
"Well, obviously, she's feeling very sad, because of Cedric dying. Then I expect she's feeling confused because she liked Cedric and now she likes Harry, and she can't work out who she likes best. Then she'll be feeling guilty, thinking it's an insult to Cedric's memory to be kissing Harry at all, and she'll be worrying about what everyone else might say about her if she starts going out with Harry. And she probably can't work out what her feelings toward Harry are anyway, because he was the one who was with Cedric when Cedric died, so that's all very mixed up and painful. Oh, and she's afraid she's going to be thrown off the Ravenclaw Quidditch team because she's been flying so badly."
We can say that Hermione was an emotionally intelligent person. She was empathetic towards Cho Chang. Also, throughout the series, she has shown multiple instances of her being able to understand and display her emotions aptly.
If we take today's present scenario into consideration, we all need to get up our EQ, since everyone is in the need of some emotional intelligence. We really can't afford to have "the emotional range of a teaspoon".
