The most common for me are verbal (if I think while taking a walk, I basically talk to myself), spatial/kinesthetic (which key on my ring goes to this door again?), and visual/symbolic (integrate Sin(Ln(x))/x dx). But there's plenty more stuff I'd call thinking.

Verbal and symbolic thinking are often just expressions of conceptual problem-solving, where my brain can actually fit concepts together rather than just talking. Like in the integral above - I pattern-match it to the general concept "change of variables," without using words or symbols, and then after a second or so of brewing I can examine the idea in terms of symbols and words.