In very broad terms (and I do mean very; there's enormous differences between individuals), the left hemisphere of the brain is used most often for logic, details, factual information, language and words, concepts of time in the sense of present and past, mathematics and science, patterns and order and the perception of patterns, reality emphasizing, knows the names of objects, strategic, and practical, and safe.
In, again, very broad terms, the right hemisphere excels at feelings, and emotions, and perceiving them, "big pictures," rather than details, imaginative and symbolic processing, present and future oriented, spatial perception, fantasy and religio-philosophical thought, knows the functions of objects, impetuous, risky,
Curious? Here's an article from Wired that suggest we're leaving the age of the left-brain and entering the age of the right-brain. And here's a page that explains brain lateralization, and it's connection to handedness, and why we would care, anyway.
Just to make it more interesting; most people, if they try, can make the dancer seem to switch spin directions.