Why elastic materials are discribed by tensors? I am starting to read about elasticity of thin surfaces and I don't understand why tensors play such a major part?
What are the tensors describing about the material?
And just to clarify - Is there some calculation often done with 2nd order tensors (describing 3d materials) that is common for elastic materials?
 A: Tensors appear when we want to relate two vectors that are proportional to each other but need not be parallel.
In order to understand where tensors appear, it may help to use the Cartesian of suffix notation for vectors. In that notation, a vector $\vec{F}$ is written as $F_i$, with $i = 1,2,3$. If a vector $\vec{G}$ is proportional to another vector $\vec{F}$ then the most general way of writing that dependence is $G_i = \tau_{ij} F_j$. This allows for the case that $\vec{G}$ is not parallel to $\vec{F}$. If we represent $G_i$ or $F_j$ as column matrices, then $\tau_{ij}$ is a square matrix connecting the two. The square matrix represents a tensor.
An example that appeals someone with a background in electrodynamics is, $\vec{D}$ is proportional to $\vec{E}$. In the case of an isotropic dielectric, $\vec{D} = \epsilon\vec{E}$, where the scalar $\epsilon$ is the material's permittivity. If the material is anisotropic, then the relation is $D_i = \epsilon_{ij}E_j$, where $\epsilon_{ij}$ is the second order permittivity tensor. Another easy to understand relation is between angular momentum and angular velocity via the moment of inertia tensor.
In continuum mechanics, we relate stress $\tau_{ij}$, a second order tensor, with strain $\gamma_{ij}$, another second order tensor. The most general relationship between them is 
$\tau_{ij} = A_{ijkl}\gamma_{kl}$, where $A_{ijkl}$ is the elasticity modulus, a fourth order tensor.
A: 'Tensor' is a general name for almost any physically-meaningful quantity and you've been working with tensors since primary school:   scalars like temperature are 0th order tensors, vectors like velocity are 1st order tensors, quantities like stress and strain are 2nd order tensors, etc.   The order of a tensor is shown by the number of subscripts its components have.   Almost any textbook with 'Continuum Mechanics' in the title will explain the details.
