What is tension in string theory? One often hears the words "string tension" in string theory. But what does it really mean? In ordinary physics "tension" in an ordinary classical string arises from the fact that there are elasticity in the string material which is a consequence of the molecular interaction (which is  electromagnetic in nature). But string theory, being the most fundamental framework to ask questions about physics (as claimed by the string theorists) can not take such elasticity for granted from the start. So my question is, what does "tension" mean in the context of string theory? Perhaps this question is foolish but please don't ignore it.
 A: A good question. The string tension actually is a tension, so you may measure it in Newtons (SI units). Recall that 1 Newton is 1 Joule per meter, and indeed, the string tension is the energy per unit length of the string.
Because the string tension is not far from the Planck tension - one Planck energy per one Planck length or $10^{52}$ Newtons or so - it is enough to shrink the string almost immediately to the shortest possible distance whenever it is possible. Unlike the piano strings, strings in string theory have a variable proper length. 
This minimum distance, as allowed by the uncertainty principle, is comparable to the Planck length or 100 times the Planck length which is still tiny (although models where it is much longer exist).
For such huge energies and velocities comparable to the speed of light, one needs to appreciate special relativity, including the $E=mc^2$ famous equation. This equation says that the string tension is also equal to the mass of a unit length of the string (times $c^2$). The string is amazingly heavy - something like $10^{35}$ kg per meter: I divided the previous figure $10^{52}$ by $10^{17}$ which is the squared speed of light.
Basic equations of perturbative string theory
More abstractly, the string tension is the coefficient in the Nambu-Goto action for the string. What is it? Well, classical physics may be defined as Nature's effort to minimize the action $S$. For a particle in special relativity,
$$ S = -m\int d\tau_{proper} $$
i.e. the action is equal to (minus) the proper length of the world line in spacetime multiplied by the mass. Note that because Nature tries to minimize it, massive particles will move along geodesics (straightest lines) in general relativity. If you expand the action in the non-relativistic limit, you get $-m\Delta t+\int dt\, mv^2/2$, where the second term is the usual kinetic part of the action in mechanics. That's because the curved lines in the Minkowski space are shorter than the straight ones.
String theory is analogously about the motion of 1-dimensional objects in the spacetime. They leave a history which looks like a 2-dimensional surface, the world sheet, which is analogous to the world line with an extra spatial dimension. The action is
$$ S_{NG} = -T\int d\tau d\sigma_{proper} $$
where the integral is supposed to represent the proper area of the world sheet in spacetime. The coefficient $T$ is the string tension. Note that it is like the previous mass (from the point-like particle case) per unit distance. It may also be interpreted as the action per unit area of the world sheet - it's the same as energy per unit length because energy is action per unit time.
At this moment, when you understand the Nambu-Goto action above, you may start to study textbooks of string theory.
Piano strings are made out of metallic atoms, unlike fundamental strings in string theory. But I would say that the most important difference is that the strings in string theory are allowed - and love - to change their proper length. However, in all the other features, piano strings and strings in string theory are much more analogous than the string theory beginners usually want to admit. In particular, the internal motion is described by equations that may be called the wave function, at least in some proper coordinates.
Also, the strings in string theory are relativistic and on a large enough piece of world sheet, the internal SO(1,1) Lorentz symmetry is preserved. That's why a string carries not only an energy density $\rho$ but also a negative pressure $p=-\rho$ in the direction along the string.
