I bet they're hard for you to imagine...if you could imagine them you'd be one of the most remarkable humans to ever live. Basically, they are no different than the three "ordinary" spatial dimensions, except they're essentially curled up into a space so tiny, about the size of the Planck length, that you traverse the entire dimension and end up back where you started (they would have to be circular) in almost no time at all so you don't notice them in our macroscopic life. If the regular dimensions are also circular, than the only difference is the fact that the higher dimensions are curled up. One major difference that you would notice if you could look into one of the extra dimensions is that since their so small, and they're circular, you would theoretically be able to see yourself. As much as I dislike referring to certain writings in posts, it seems almost like a shortcut, Brian Greene explains them much better than I could in his book "The Elegant Universe". Their existence was first proposed by Franz Kaluza early in the 20th century. He wrote Einstein, who immediately expressed interest in the idea, only to express skepticism soon after. However, he eventually came around and offered to present Kaluza's paper to the Academy. The fascinating aspect of extra spatial dimensions is that if general relativity is applied to them, just as gravity can be explained as disturbances in the familiar, extended, three dimensions, the other fundamental forces can be explained as disturbances in the extra dimensions. When Kaluza added a fifth dimension to Einstein's equations, it was Maxwell's equations of electromagnetism that emerged! Unfortunately, there were some unresolved problems with extra dimensions, so the idea was put on the shelf. When string theory came around, it was revived again. In case you aren't familiar, string theory (the main focus of "The Elegant Universe") is built around the premise of all fundamental and messenger particles being just vibrating loops of energy. It's an attempt at a theory of everything, trying to unify the four fundamental forces as well as marry the conflicting general theory of relativity with quantum mechanics. One problem it was experiencing was that the incomplete calculations of string theory were producing negative probabilities. By increasing the number of dimensions a string could vibrate in, the negative numbers started to decrease. In fact, by increasing the number of spatial dimensions to nine, the negative numbers dissappeared altogether. Due to the incomplete nature of string theory, a more robust form of it called M-theory suggests that the number may actually be ten spatial dimensions, and one time dimension, for a total of eleven. Of course string theory still has a long way to go, and is still speculation for the most part. However the idea is intriguing, and extra dimensions do provide a very complete picture of reality. To answer your final question, yes a 6-dimensional is possible, actually if M-theory is correct than every object in the universe has ten spatial dimensions, since it's fundamental strings are 10-dimensional. However, since the curled up dimensions are so tiny we don't notice our extended nature. Some work by Einstein suggests that in theory curled up dimensions could be extended...wouldn't that be interesting. Firing up the hyperdrive, captain...
