The Palatini action is a way to formulate

general relativity in

Lagrangian form. It is a bit like the Einstein-Hilbert action, except that it is not written in terms of the Riemann curvature

Tensor, but rather in terms of a "Lorentzian" curvature tensor F and a "frame field" e, as follows:

L_p = F_alpha_beta^I^J e_I^alpha e_J^beta sqrt(det g_alpha_beta)

where g_alpha_beta is the usual metric tensor. The frame field e_I^alpha has the property that

g_alpha_beta = eta_I_J e^I_alpha e^J_beta

where eta_I_J is the ordinary Minkowski metric.
This means that F measures the curvature in a frame, where the metric is locally flat. This is useful in quantum gravity in some way, but I am not quite sure how.
Anyway, like the ordinary Riemann curvature tensor can be formed from the Christoffel symbols, the F can be formed from a Lorentzian connection, which is a Levi-Civita connection in a locally flat frame. With one more trick, called "self-duality", which I should write more about, this Lagrangian can be canonically quantized. It turns out that the frame field represents the canonical momentum conjugate to the Lorentzian connection. Using self-duality it is also possible to formulate the constraints so that they are closed under Poisson brackets.

Many supergravity theories also employ Palatini-type formalism. I'll write more when I learn more about it.