The Nambu-Goto action is an action functional for sigma-models with target space a (pseudo) Riemannian manifold $(X,g)$: it is the induced volume functional
where $dvol(\gamma^* g)$ is the volume form of the pullback $\gamma^* g$ of the metric tensor from $X$ to $\Sigma$.
Let
$p \in \mathbb{N}$ (for p-brane dynamics);
$(X,g)$ a pseudo-Riemannian manifold (target spacetime);
$\Sigma$ a compact smooth manifold of dimension $(p+1)$ (worldvolume).
$[\Sigma,X]$ the diffeological space of smooth functions $\Sigma \to X$.
For $\phi \colon \Sigma \longrightarrow X$ the induced “proper volume” or Nambu-Goto actionof $\phi$ is the integral over $\phi$ of the volume form of the pullback of the target space metric $g$ to $\Sigma$.
Notice that the rank-2 tensor $\phi^\ast g\in \Gamma(T* \Sigma \oplus T* \Sigma)$ is in general not degenerate (unless $\phi$ is an embedding), hence is in general not, strictly speaking a pseudo-Riemannian metric on $\Sigma$, but nevertheless it induces a volume form by the standard formula, only that this allowed to vanish pointwise (and even globally, for instance if $\phi$ is constant on a single point). In the literature $dvol(\phi^\ast g)$ is usually written as $\sqrt{-g}d^{p+1}\sigma$.
The NG is classically equivalent to the Polyakov action with “worldvolume cosmological constant”. See at Polyakov action – Relation to Nambu-Goto action.
The NG-action serves as the kinetic action functional of the sigma-model that described a fundamental brane propagating on $X$. For $dim \Sigma = 1$ this is the relativistic particle, for $dim \Sigma = 2$ the string, for $dim \Sigma = 3$ the membrane.
The Nambu-Goto action functional is named after Yoichiro Nambu.
Detailed discussion of the relation to the Polyakov action and the Dirac-Born-Infeld action is in
One string theory textbook that deals with the Nambu-Goto action in a bit more detail than usual is