Genius, as I personally define the term, refers to an individual with great multitudinal prowess. He can think not only in the present, but beyond what the conventions dictate. He is by far beyond measurable by current standards. To think that he is merely what he seems to be based on his present outputs is an underestimation. Likewise, he can also see what other may have not seen before by his contemporaries.
So, how do geniuses think? To think like a genius, you must be one of them.
I think geniuses can be superficially assessed as being ordinarily adroit in everything they do. They may be considered jack-of-all trades. When they think and apply it, they ooze with confidence in what they do. Whether by objective or intersubjective standards and norms, even lay people would concur that indeed that geniuses are 'completing' a magnum opus even when it is simply not complete yet.
Geniuses, in other words, based on the latter paragraph think on their own terms and standards. They can do an incomplete project and yet can be hailed already as a masterpiece. Even more highly profound to the mediocre is when a genius has fully realized the figment of his imagination in tangible ways. See the super-engineering projects, modern architectures, arts, inventions, and so on? They are achievements of geniuses made realizable by people tended by such individuals.
We could possibly list here a few of the individuals who were considered by most people as geniuses. Though and an exhaustive enumeration and not in order of their IQs, achievements or popularity, we have, viz.:
Bernhard Riemann (1826-1866)
Leonardo da Vinci (1452-1519)
Marie Skłodowska-Curie (1867-1934)
Albert Einstein (1879-1955)
Nikola Tesla (1856-1943)
Gottfried Leibniz (1646-1716)
William Shakespeare (1564-1616)
Terence Tao (1975-____)
A few individuals may not consider all of the abovementioned people as geniuses because some believe that geniuses are a rare breed of being jack-of-all trade. However, for my part, geniuses are persons who have contributed significantly well to merit praise in human way. They do not have to necessarily become immortalized by a few pages of comments and writings. Their legacies say it all.
I also do not believe that a person was never born a genius. We are also geniuses in our own little or big ways. There are simply many factors why some peoples' geniuses have never come to fruition. To delimit the meaning and essence of a genius is an underrating. I know that deep within us all there is a genius still awaiting to manifest. One way or the other, we can be one. The question is really: when? for there is only a limited time that we have to live and afterwards die.
So, how about the question about how geniuses think most of the time?
Well, they think mostly in innovative, critical and creative manners. They think of things that can better harness and tap their talents, focus, and novel thinking. If we only can do the same in our spare time most of the time, after some time, we can also be considered like them...