How many dimensions are there and how come I can only move through three?
Loosely speaking, a dimension of space is the minimum number of independent coordinates needed to specify each point within the space. In the context of our physical world, we can describe every event by 3 spatial dimensions and one dimension of time (an event occurred in that location, and at that time). Thus, we say that our world is composed of "3+1" dimensions. Clearly, the time dimension is separated from the space dimensions, as there is only one of it, and as we cannot move freely in it, but are forced to move only in one direction - towards the future.
We do not have any observational or experimental evidence for the existence of any other dimensions beyond our "3+1" dimensional space. Nonetheless, some modern physics theories, such as superstring theory, that attempts to unify the four fundamental forces of nature, suggests that there are in fact more dimensions (10 in this particular theory). Since we do not observe these dimensions, clearly, if they do exist there must be some mechanism that hides them from us. One suggestion is that these extra dimensions may be "curled up" at such tiny scales as to be effectively invisible to current experiments.
To my knowledge, there is no really convincing answer to the question of why there are only 3 "infinite" spatial dimensions, while other dimensions (if exist) are "curled up". One suggestion is that this is related to the expansion of the universe, originating from the big bang. The idea is that before the big bang, the cosmos was perfectly symmetrical, 9 dimensional universe. However, the universe was unstable, and thus spontaneously broke, "cutting" the dimensions into two separate space-times: the (3+1) universe that we live in that is rapidly expanding, and another 6 dimensional one that contracted violently and shrank. Again, this is a highly speculative idea, as clearly we have got no observational clue on what happened before the big bang, and similarly there is no observational evidence that can support the idea that there are compact extra dimensions at all.