I see it as a sort of collection of skills of critical thinking, analysis, awareness, empathy, pattern matching, inductive reasoning, intuitive logical leaps, etc. And all the skills exist on a bell curve. You have some people at one end who will never be good at various skills, no matter what. You have some people at the other end who will always be good at those skills no matter what. You have most people in the middle where they will be good or excellent at various skills through a combination of innate talent and hard work.
And people who can make up for one skill with another. I know people who just don't intuitively grasp emotional responses. But, they want to be empathetic and helpful, so use logical analytical skills to learn how to ask questions and create a sort of mental database of reactions. They "hack" the emotional intelligence deficiency by turning into an analytic problem, which they are good at. And, it works quite well, actually.
That's a really cool thing, because it means that a person can get better at any of these skills. And it means that people with ADHD, dyslexia, dyscalculia, or other learning differences can learn how to use their strengths to help compensate for the weaknesses. There's a lot of research to back this up, too.
I don't think there is one intelligence skill that is more or less important that another. I get frustrated with the people who are dismissive of emotional intelligence. I also get frustrated with the people who dismiss academic achievement. Its all important, its all a skill, and its all something someone can be proud of having done. And, of course, not being good at an intelligence skill says absolutely nothing about the worth of the person in question. Being intelligence doesn't make anyone better than anyone else. Being unintelligent doesn't make anyone else less than someone else.