How many dimensions there are?

Depends what you mean by "dimension", since there are several definitions of the word. In mathematics, there are an infinite number of dimensions, since (for the most part, except in geometry) dimension simply refers to an array of numbers or co-ordinates. It is therefore possible to construct a graph (or build a co-ordinate system) with any number of "dimensions". Many branches of mathematics deal with these higher co-ordinate structures. In geometry, dimension has a similar definition to the physical sense, but again an infinite number of dimensions are available, since these do not necessarily refer to "real world" applications. Euclidean geometry deals with 2-dimensional spaces, while non-Euclidean geometries deal with 3, 4 or more "dimensions" of object.

In physics (the sense I think you mean) the question is still somewhat open. Originally we believed there were only three, but Einstein demonstrated that time was another dimension, and thus established that there were four (three spatial and one time). Recently, certain problems in physics have suggested that even "higher" dimensions exist. These dimensions are purely (as far as we know) spatial, but are "wrapped up" tightly around themselves, and are thus invisible and undetectable in most situations.

Depending on which theory you go for, there could be five, eight, or even eleven dimensions out there. The current strongest candidate (but far from proven yet) is M-Theory, which holds that there are 11 spatial dimensions, wrapped up into what are called Calabi-Yau manifolds, like little knots at every point in space-time.

These dimensions will never be "viewed" by human eyes, even if proven or otherwise detected, since our entire sensory system is built around 3 perceivable spatial dimensions, and thus we would have no frame of reference to understand what we would be seeing.
We operate in three dimensions, plus time. That's four. But there can be more. In mathematics, any number of dimensions can be managed (or attempted, at least). Theoretical physicists are currently working with a dozen or so in what are called manifolds in an attempt to understand reality as we know it. The number of dimensions will vary as the individual who is manipulating them or working in them. Usually the x, y, z axes and time are sufficient for most of us.
This is an interesting question in contemporary physics. Albert Einstein described the universe as existing in 4 dimensional space-time, but M-theory (an extension of string theory) postulates that there are 11 space-time dimensions.

(P.S. How many dimensions are there?)
One. Unless in the future humans create a device which can create others.