As is usually done when first presenting string theory, the Nambu-Goto Action,
$$
S_{\text{NG}}:=-T\int d\tau d\sigma \sqrt{-g}
$$
($g:=\det (g_{\alpha \beta})$ is the induced metric on the world-sheet and $T$ is a positive real number interpreted as the string tension), is introduced as the natural generalization of action for a relativistic point particle, which in turn is obviously a correct action as it produces the proper equations of motion (and has a nice geometric interpretation).
Not long after the introduction of the Nambu-Goto action, authors tend to introduce the Polyakov action,
\begin{align*}
S_{\text{P}} & :=-\frac{T}{2}\int d\tau d\sigma \, \sqrt{-h}h^{\alpha \beta}g_{\alpha \beta}=-\frac{T}{2}\int d\tau d\sigma \, \sqrt{-h}h^{\alpha \beta}\partial _\alpha X\cdot \partial _\beta X \\
& =-\frac{T}{2}\int d\tau d\sigma \, \sqrt{-h}h^{\alpha \beta}\partial _\alpha X^\kappa \partial _\beta X^\lambda G_{\kappa \lambda}(X),
\end{align*}
where $G_{\kappa \lambda}$ is the space-time metric, $g_{\alpha \beta}$ is the induced metric on the world-sheet, and $h_{\alpha \beta}$ is the auxiliary metric on the world-sheet ($h:=\det (h_{\alpha \beta})$). They then usually proceed to show that these two actions are equivalent, in the sense that you can deduce the equations of motion for $S_{\text{NG}}$ given the equations of motion for $S_{\text{P}}$.
Now, that's all well and dandy, but that doesn't exactly show how one would actually arrive at the Polyakov action. You can't make it as a theoretical physicist by mindlessly computing things to show you get the right answer; you have to be able to actually, you know, come up with things. Hence, instead of just pulling the Polyakov action out of a hat, it would be nice to know a way of deriving or motivating the action.
So then, imagine you are handed $S_{\text{NG}}$ and you set out to come up with an equivalent action that, at the very least, doesn't involve a square-root. How do you come up with the Polyakov action?
Best Answer
I) The closest cosmetic resemblance between the Nambu-Goto action and the Polyakov action is achieved if we write them as
$$\tag{1} S_{NG}~=~ -\frac{T_0}{c} \int d^2{\rm vol} ~\det(M)^{\frac{1}{2}} , $$
and
$$\tag{2} S_{P}~=~ -\frac{T_0}{c}\int d^2{\rm vol}~ \frac{{\rm tr}(M)}{2} , $$
respectively. Here $h_{ab}$ is an auxiliary world-sheet (WS) metric of Lorentzian signature $(-,+)$, i.e. minus in the temporal WS direction;
$$\tag{3} d^2{\rm vol}~:=~\sqrt{-h}~d\tau \wedge d\sigma$$
is a diffeomorphism-invariant WS volume-form (an area actually);
$$\tag{4} M^{a}{}_{c}~:=~(h^{-1})^{ab}\gamma_{bc} $$
is a mixed tensor; and
$$\tag{5} \gamma_{ab}~:=~(X^{\ast}G)_{ab}~:=~\partial_a X^{\mu} ~\partial_b X^{\nu}~ G_{\mu\nu}(X) $$
is the induced WS metric via pull-back of the target space (TS) metric $G_{\mu\nu}$ with Lorentzian signature $(-,+, \ldots, +)$.
Note that the Nambu-Goto action (1) does actually not depend on the auxiliary WS metric $h_{ab}$ at all, while the Polyakov action (2) does.
II) As is well-known, varying the Polyakov action (2) wrt. the WS metric $h_{ab}$ leads to that the $2\times 2$ matrix
$$\tag{6} M^{a}{}_{b}~\approx~\frac{{\rm tr}(M)}{2} \delta^a_b~\propto~\delta^a_b $$
must be proportional to the $2\times 2$ unit matrix on-shell. This implies that
$$\tag{7} \det(M)^{\frac{1}{2}} ~\approx~ \frac{{\rm tr}(M)}{2},$$
so that the two actions (1) and (2) coincide on-shell, see e.g. the Wikipedia page. (Here the $\approx$ symbol means equality modulo eom.)
III) Now, let us imagine that we only know the Nambu-Goto action (1) and not the Polyakov action (2). The only diffeomorphism-invariant combinations of the matrix $M^{a}{}_{b}$ are the determinant $\det(M)$, the trace ${\rm tr}(M)$, and functions thereof.
If furthermore the TS metric $G_{\mu\nu}$ is dimensionful, and we demand that the action is linear in that dimension, this leads us to consider action terms of the form
$$\tag{8} S~=~ -\frac{T_0}{c}\int d^2{\rm vol}~ \det(M)^{\frac{p}{2}} \left(\frac{{\rm tr}(M)}{2}\right)^{1-p} , $$
where $p\in \mathbb{R}$ is a real power. Alternatively, Weyl invariance leads us to consider the action (8). Obviously, the Polyakov action (2) (corresponding to $p=0$) is not far away if we would like simple integer powers in our action.