I think there are two types of genius, Creative and Logical. The greatest geniuses, the Supergeniuses, can merge the two in moments of clarity.
THE CREATIVE GENIUS
I posit that Creative geniuses do most of their thinking in their unconscious. While Logical geniuses do most of their thinking in their conscious. To make a crude example, consider an alphabet soup with 50 letters. The unconscious mind does not see these as letters, does not see an order to these letters, does not even recognize shape in these letters. It see 50 objects and 100 spaces. It arranges these 50 objects in these 100 spaces at random, with no semblance of logic or rationalle attached. 9/10 times the unconscious spells complete gibrish. 1/10 times the unconscious spells 'eureka' surrounded by 44 random letters. The problem is the unconscious, even having spelled 'eureka', cannot recognize this word because it does not even know what letters are. The Supergenius has the ability to consciously see and read this 'eurkea' and can thus bring it into the public domain of language. The creative genius only realizes the beauty in the arrangement of these 50 letters, but is unaware of its logical significance.
THE LOGICAL GENIUS
Again, we think of an alphabet soup with 50 letters and 100 spaces. The logical genius lives largely in his/her own conscious. He sees letters and he sees words. He only assembles the letters into words of semblance. He is cumbersome in his task and he is limited to his predefined vocabulary of words. He may eventually spell 'eureka', but it takes him much time and effort to tediously swap letters in and out of allowable positions. His Creative genius friend simply gives the soup a swirl and let's the letters coalesce as they may. The logical genius cannot afford such a luxury and it takes him much longer to complete an entire swirl.
THE SUPERGENIUS
I believe the supergenius is unique. I speculate that his neural pathways have extra links between his unconscious domain and conscious domain. He can swirl the letters quickly and at the same time, each swirl produces a series of cogent, tangible words. His unconscious swirls at such a fast rate and his conscious mind samples 'eurekas' so adeptly from the soup that he is litterally bombarded with revelations at most instants of the day.
If you have ever had a breakthrough in your area of discipline, or have noticed that you randomly had an insight into an area you weren't even thinking about, I think you have experienced a glimpse into this world.
Its interesting to note that Einstein said that many of his major breakthroughs came to him in dream. He frequently would pour over complex math papers for hours on end, then he would drift into a mid-day nap, whereupon he'd wake up suddenly with a 'eureka' type answer to his nagging question. He even stated that this works best in daytime when your sleep is very light. When you are barely unconscious and your conscious mind is free to mingle with your unconscious' free associations.
This is sort of the reverse of pschizofrenics (spelled wrongly I'm sure, lol). They are victimized by their unconscious dipping into their conscious, while awake. Supergeniuses have the ability to dip their conscious into their unconscious and they can even take a teaspoon of it back into the world of the public domain of language, so that you and I can equally taste its sweetness.
This is my opinion, but I think its a pretty good one.