(This post was originally published in late 2009. To my surprise, it's received some attention from random people throughout the Internet. I've refreshed it a little and posted here, hoping it has retained its relevance)
To me, conceptual thinking is simply the ability to effortlessly walk up and down the ladder of abstraction.
Most people believe that great conceptual thinkers simply think in more abstract terms than non-conceptual thinkers. The ability to think abstractly (i.e. in terms of themes, patterns, and generalizations, rather than specifics, examples, and concretes) is a necessary condition, but I don't think it is sufficient .
While someone who is just an abstract thinker is likely able to solve problems more easily (by separating themselves from distracting details and focusing on the core of the problem), their impact is limited. For example, they find it hard to communicate with those unable to think abstractly (the majority of the population!) so they limit the extent of their influence and collaboration. They also often struggle to shape reality, because, like it or not, reality is concrete and not abstract.
To make their ability useful, abstract thinkers need to be able to convert an abstract thought into a concrete one, and vice versa. This ability is what I call conceptual thinking. A conceptual thinker (comfortably) starts in the concrete, then walks up the hierarchy of abstractions. At some level they make connections between the abstract representation of the concrete thought and another abstract representation. If need be, they can then walk that abstract thought back into another, very different concrete thought. The idea is that a local search (i.e. making connections) in the abstract space is easier than a local search in the concrete space. But by starting and ending in the concrete, that person can both communicate more effectively with others, and actually make things come true.
Suppose I have to solve a difficult puzzle. If I am a conceptual thinker, I can turn the problem into a more abstract one (recognize it as an instance of a more generic class of problems). The "abstraction tree" is sparser the higher you go, because each time you walk "up" the hierarchy, you collapse a number of more concrete thoughts into one more abstract one. At some point, you're likely to have encountered an abstraction that you have seen before, find the answer you're looking for, and then walk the ladder all the way "down" by adding layers of concreteness to actually find the solution to the specific puzzle. That way, instead of solving hundreds of puzzles in hope that in the future you will encounter one, you really just need to solve one.
Unfortunately, this ability to walk up and down the ladder of abstraction is not easy. You replace the ability to solve a concrete problem with the ability to turn concrete things into abstract things, and vice versa. In the latter, the higher the abstraction tree you go, the harder the walk becomes.