For most intents and purposes, there are 4. The three spatial dimensions, and time.
However, some current postulates require the existance of other dimensions, as part of what is called a Calibi-Yau space. These are usually referred to as being 'curled up', as they're pretty damn small.
As an example, consider an ant on a rope. If it goes forwards or backwards, it can (assuming an infinite rope) go on forever. If it goes left or right, it can only go a short distance before coming back on itself, as it walks around the rope. The curled up dimensions are like going round the outside of the rope.
Of course, that's only speculation at the current point in time.