I am reading 'A first course in string theory' by Barton Zwiebach (2ed) on page 112 he comes up (after a small derivation) the action formula:
$$S=-\frac{T_0}{c} \int d\tau d \sigma \sqrt{-\gamma}.$$
In doing so he makes a few assumptions and does not explain why we would expect the action as given here to obey the Hamilton's principle of stationary action?
Also is this action unique or could we have formulated it differently?
I) The action principle of a theory is the usually taken as the first principle of a theory, and therefore it can strictly speaking not be derived. Nevertheless, the Nambu-Goto action is a natural a generalization of the following line of thought:
In a Riemannian space $(M,g)$ [with Euclidean signature], a geodesics are (locally) the shortest path between points. We therefore have a variational principle by choosing the action to be the length of the curve $$\tag{1} S~=~ \int_a^b\! d\lambda ~\sqrt{g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}.$$
The action for a relativistic point particle in a curved spacetime is the natural generalization of (1). [Note however, that the Minkowski signature of the spacetime metric leads to the important new caveat that we have to assume that $\dot{x}^{\mu}$ is not spacelike, so that the argument of the square-root remains non-negative]. The action is now proportional to the proper time of the point particle, which is the natural invariant notion of length of a world-line in spacetime.
It is hence natural to expect that the Euclidean action for a string should be the area of the world-sheet.
The Nambu-Goto action is the equivalent of pt. 3 if the target space metric has Minkowski signature.
II) Generally speaking, an action principle induces a set of Euler-Lagrange eqs. It is often possible to find equivalent actions that yields the same Euler-Lagrange eqs. E.g. the Polyakov action and Nambu-Goto action are classically equivalent, as 0celo7 mentions.
The action principle holds by assumption. It is assumed that all equations of motion follow from this principle with the appropriate action.
By introducing an auxiliary tensor field $h_{\alpha\beta}$, one may write down the so-called Polyakov action $$S_\mathrm{Poly}=-\frac{T}{2}\int_\Sigma \mathrm{d}^2\sigma\,\sqrt{-\operatorname{det}h}h^{\alpha\beta}X^\mu_{,\alpha}X^\nu_{,\beta}g_{\mu\nu}$$ which is easier to work with. Using the Euler-Lagrange equations for $h_{\alpha\beta}$, one may verify it is equivalent to the NG action on-shell.
We are assuming that this is the case. In general, one can think that every Lagrangian corresponds to some equations of motion, so the NG action corresponds to some sort of motion of a string.
The powerful justification for this action is that it is manifestly Lorentz covariant, and one can show that the equations of motion that it leads to are those one would expect of a relativistic string, such as the wave equations describing the oscillations in each dimension.
