We say that a submanifold $M$ of a Riemannian manifold $(N,h)$ is minimal if it's second fundamental form is traceless i.e (as far as I understand)
Trace $B = \sum_{0}^{m}B(X_{i},X_{i})=0$,
for any orthonormal frame for the tangent bundle $TM$.
Is this in some sense multidimensional analogue of a geodesic? Minimal surfaces seams to be the minimize the area functional instead of length functional.
Yes, indeed. The trick is to realize that geodesics in a Riemannian manifold $N$ are critical points (in a suitable sense, see below) for the energy functional on the space of maps ${\mathbb R}\to N$. Now, to extend this interpretation in higher dimensions, fix two Riemannian manifolds $M, N$ and consider the space of Riemannian isometric immersions $Imm(M,N)$. Since $M$ need not be compact one has to restrict energy variation to variations which are trivial away from compacts (i.e. are "compactly supported"). Then $f\in Imm(M,N)$ is minimal iff it is harmonic iff it is a critical point of energy functional under compactly supported variations. See
J.Eells and J.Sampson, Harmonic mappings of Riemannian manifolds, American Journal of Mathematics, Vol. 86, No. 1 (1964), pp. 109-160.
This is not the end of the story since harmonic and minimal maps (unlike nonconstant geodesics) need not be immersions, but I think, this suffices for your purposes.