Well we can only know about the ones below ours. However, the 2nd, 1st and 0th dimension are concepts we have created ourselves... they function simply to better our understanding of the universe. But the dimensions above us are complicated. They have been theorised much in the same way we would fill in a sequence... If one man says 1, 2, 4, 8... naturally we continue with 16, 32, 64... but that might not necessarily be correct. It is the most logical. However, the sequence may have just been that the next number is greater than the previous one. The links between them may not be relevant. It has been theorised that there are only 10 dimensions. As Rob Bryanton says, a being from the second dimension visiting the third dimension will only see things in 2-dimensional cross-sections. For example... a balloon to a 2-dimensional being will appear as a single point that gets larger and larger and then starts to get smaller and smaller and then disappears out of existence. This concept has been used to explain the 4th dimension. If we were to live in a 4th dimensional world (known as time), where beings can look behind them and see themselves as a child and also look in front where they see themselves as an old man and then ultimately a corpse, then we unfortunately would only be able to see in 3-dimensional cross-sections of time. We would only see a brief moment of each section in time as it progresses. We only see "the present". It's a complicated theory and to explain it in this form would take too long. Also reading Rob Bryanton's enthralling book might take a bit too long aswell but it really goes into the idea in depth. But there is a short presentation that explains the theory found on this website "
http://www.tenthdimension.com/".