Why does string theory require 9 dimensions of space and one dimension of time? String theorists say that there are many more dimensions out there, but they are too small to be detected. 


*

*However, I do not understand why there are ten dimensions and not just any other number? 

*Also, if all the other dimensions are so coiled up in such a tiny space, how do we distinguish one dimension from the other? 

*If so, how do we define dimension?   
 A: Let me take parts 2. and 3. of the question first:
The 10 dimensions of string theory are, a priori, not "coiled up" or anything else. They are derived for a string theory where the classical version of the string propagates in d-1 spatial dimensions and 1 temporal dimension, i.e. Minkowski space $\mathbb{R}^{1,d-1}$. "Dimension" here is dimension of a manifold in the usual sense of differential geometry - number of coordinates needed to uniquely distinguish a point on the manifold from all points close to it.
Now, as for why (super)string theory in flat space requires $d=10$:
One way to see string theory is by certain two-dimensional conformal field theories living on the world sheet the string traces out in the target space. I give a quick explanation of the structure of such theories here. The total conformal charge of the full combined CFT on the worldsheet can be seen as the quantum anomaly of the classical Weyl symmetry of the string - for a general discussion of the relation between anomalies and central charges see this answer by DavidBarMoshe, for a general discussion of the relation between central charges and quantization see this Q&A of mine.
The quantization of the bosonic (or "naive") string has d coordinate fields that each correspond to a free bosonic CFT with central charge $c=1$ plus a "ghost system" incurred from BRST quantization that has a central charge $c=-26$. Ghost systems are allowed to have negative central charge because they decouple from all physical processes.
Now, the procedure used to quantize this string in the first place makes use of the Weyl symmetry being non-anamalous, i.e. $c=0$ for the full theory - which only happens at $d\cdot 1 - 26 = 0$, i.e. $d=26$. Therefore, the bosonic string exists consistently as a quantum theory only in 26 dimensions.
The superstring is now what you get when you additionally have fermions living on the worldsheet. It's called the "super"string because the new action is supersymmetric, but it might as well be called the "spinning string", since trying to write down a worldline action for a particle with spin also introduces such fermions.
In any case, the ghost system for the larger symmetry of the superstring has $c=-15$, and the fermions each contribute $c=1/2$. This gives the requirements $\frac{3}{2}d - 15 = 0$, which is solved by $d=10$.
I'm afraid the full derivation is rather technical and it would serve little use to reproduce it here. Lastly, one should remark that there are many equivalent ways to arrive at this constraint on dimensions, this is by far not the only one, but the one that's easiest to tell for me. Others might find a presentation discussing ordering constants related to the vacuum energy more physically intuitive, for example.
A: (1) String Theory is a very mathematical theory based on some natural assumptions, and this ends up relating Quantum Mechanics and General Relativity, as we want. Some of the equations in String Theory, however, have a proportionality constant $c$ in it, called the central charge. And when we manipulate these equations and set them equal to each other, we see that they ONLY make sense if $c=26$. This $c$ is the dimension of space that String Theory is a priori defined over, so now we see that we need 26 dimensions to not have absurdities... BUT that only made use of the bosonic particles in the world -- we forgot about fermions!! This is where Supersymmetry comes into play, and it throws in the fermions, and the equations are perturbed and leads to a new dimension of 10 for everything to make sense.
(2) Just because we can't see it, doesn't mean it's not there... we can't see atoms with the eye, but we can use tools to see them... same thing happens here, our current technology can't see them, but we hope to change this in the future. EVEN BETTER though, is that the formula for gravitational force should actually be different because of these extra small dimensions -- thus we plan to figure these extra dimensions out by testing the gravitational force at small distances and seeing a perturbation to the standard inverse-square law of Newton. These extra dimensions are what is supposed to make gravity so weak compared to the other forces of nature.
(3) a dimension is just a coordinate axis... so time is a dimension too. And just like your clock, this axis can repeat itself and not stretch to infinity.
A: For bosonic string theory, see this. I'll be using the same standard notation in this answer.
Superstrings (in the RNS formalism)
Ramond sector
\begin{array}{l}0 = {{\hat G}_0}\left| \psi  \right\rangle \\{\rm{ }} = \sum\limits_{n =  - \infty }^\infty  {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n}} \left| \psi  \right\rangle {\rm{                             }}\\{\rm{ }} = \left( {{{\hat \alpha }_0}\cdot\,{{\hat d}_0} + \sum\limits_{n = 1}^\infty  {\left( {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n} + {{\hat d}_{ - n}}\cdot\;\;{{\hat \alpha }_n}} \right)} } \right)\left| \psi  \right\rangle {\rm{                                      }}{\kern 1pt} \,\\{\rm{ }} = \left( {\left( {\frac{1}{2}{\ell _P}{p^\mu }} \right)\,\cdot\,\left( {\frac{1}{{\sqrt 2 }}{\gamma ^\mu }} \right) + \sum\limits_{n = 1}^\infty  {\left( {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n} + {{\hat d}_{ - n}}\cdot\;\;{{\hat \alpha }_n}} \right)} } \right)\left| \psi  \right\rangle \\{\rm{ }} = \left( {\frac{1}{{2\sqrt 2 }}{\ell _P}{\gamma ^\mu }{p_\mu } + \sum\limits_{n = 1}^\infty  {\left( {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n} + {{\hat d}_{ - n}}\cdot\;\;{{\hat \alpha }_n}} \right)} } \right)\left| \psi  \right\rangle \left( {} \right)\\{\rm{ }} = \left( {\frac{1}{{2\sqrt 2 }}{\ell _P}{\gamma ^\mu }{p_\mu } + \sum\limits_{n = 1}^\infty  {\left( {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n} + {{\hat d}_{ - n}}\cdot\;\;{{\hat \alpha }_n}} \right)} } \right)\left| \psi  \right\rangle \\\left( {\frac{1}{{2\sqrt 2 }}{\ell _P}{\gamma ^\mu }{p_\mu } + \sum\limits_{n = 1}^\infty  {\left( {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n} + {{\hat d}_{ - n}}\cdot\;\;{{\hat \alpha }_n}} \right)} } \right)\left| \psi  \right\rangle  = 0\\\left( {{\gamma ^\mu }{p_\mu } + \frac{{2\sqrt 2 }}{{{\ell _P}}}\sum\limits_{n = 1}^\infty  {\left( {{{\hat \alpha }_{ - n}}\cdot\;\;{{\hat d}_n} + {{\hat d}_{ - n}}\cdot\;\;{{\hat \alpha }_n}} \right)} } \right)\left| \psi  \right\rangle  = 0\end{array}
This is the Dirac-Ramond Equation.
Still in the Ramond sector,
$${\hat L_0}\left| \psi  \right\rangle  = \hat G_0^2\left| \psi  \right\rangle $$=
$${\hat L_0}\left| \psi  \right\rangle  = \hat G_0^2\left| \psi  \right\rangle $$
$$a = 0$$
Now, consider some Level 1 Neveu-Schwarz Spurious State Vector $\left| \varphi  \right\rangle  = {\hat G_{ - 1/2}}\left| \chi  \right\rangle $
$$0 = {\hat G_{1/2}}\left| \chi  \right\rangle  = {\hat G_{3/2}}\left| \chi  \right\rangle  = \left( {{{\hat L}_0} - a + \frac{1}{2}} \right)\left| \chi  \right\rangle $$
So, $a = \frac{1}{2}$ in the Neveu - Schwarz sector.
Now, we consider a Ramond Spurious State Vector $\left| \varphi  \right\rangle  = {\hat G_0}{\hat G_{ - 1}}\left| \chi  \right\rangle $ ; where ${\hat F_1}\left| \chi  \right\rangle  = \left( {{{\hat L}_0} + 1} \right)\left| \chi  \right\rangle  = 0$
$$0 = {\hat L_1}\left| \psi  \right\rangle  = \left( {\frac{{{{\hat G}_1}}}{2} + {{\hat G}_0}{{\hat L}_1}} \right){\hat G_{ - 1}}\left| \chi  \right\rangle  = \frac{{D - 10}}{4}\left| \chi  \right\rangle $$
Thus, $D=10$.
A: One can posit mathematical string theories in any dimensions of any kind.

However, I do not understand why there are ten dimensions and not just any other number?

The specific dimensions arise from the requirements of the known physics encapsulated in the Standard Model and other data coming from  particle physics,  plus the requirement of General Relativity and its quantization. The Special Unitary groups whose representations accommodate the SM need at least these dimensions. There are models with more dimensions than this.

Also, if all the other dimensions are so coiled up in such a tiny space, how do we distinguish one dimension from the other?

We cannot move into the coiled ones, only in $x,y,z$.  We do not need to distinguish them, as we do not distinguish the molecules in the air. The predictions from this type of theory on the behavior of particles is the only way of checking for their existence: consistency of theory with data. 

If so, how do we define dimension? 

A space variable ( centimeters) or time one ( seconds)  that is continuous and maps the real numbers, each dimension at $90^{\circ}$ to the rest, an extension of how we define normal $x,y,z$.That some are curled should not bother one. The coordinates over the earth are curled over the sphere's surface, for example, the $90^{\circ}$ does not hold there. It would hold on the surface of a cylinder , $z$ from $-\infty$ to $\infty$, $x$ from $0$ to $2\pi r$.
A: To make the math work.  Ever since Einstein determined that time is actually another dimension, Physicists have used that notion to expand the conception of the Universe to include added (by not sensible) dimensions to get their math and theories to work.  Of particular note is Witten's unification of string theories which "only" required the addition of yet another dimension.
What string theorists fail to realize is that each dimension represents an added degree of freedom and thus may very well be under constraining the system.
In part because of this, that is why some critics of string theory (Woit, Smolin) have called string theory "Not Even Wrong" and "The Theory not of 'Everything' but 'Anything' ".
