Tensor is a vector generalized to 3Dimensions. Tensors are typically found in structural engineering for instance with anisotropic materials with characteristics which change in direction. Here you can see a typical strain tensor
Try thinking of a vector but with two dimensions that is written down in matrix form. For example the strain tensor in a solid like Viscousflow mentioned. Or the stress tensor in a fluid. The the components of the stress tensor are
Txx Txy Txz
Tyx Tyy Tyz
Tzx Tzy Tzz
So for an element in the fluid Txx would be the stress on the x face in the x direction. So it is a normal stress or pressure. Txy would be the stress on the x face in the y direction, shear stress caused by viscosity.