tensor
A scalar is one number that does not care which way you point your axes — temperature, mass, charge. A vector is an arrow that exists before you choose axes; pick a frame and it shows you components. A tensor is the natural continuation of this idea: a single geometric object that exists independently of coordinates, but which, once a frame is chosen, presents itself as an array of components carrying several indices. Scalars are tensors of rank 0, vectors and covectors are rank 1, the metric and the stress tensor are rank 2, and so on.
Precisely, a tensor of type (p, q) is a multilinear machine: it has p upper slots and q lower slots, and it takes in q vectors and p covectors and returns a number, linearly in each input. Its components carry p upper and q lower indices, like T^{ij}_k. The defining property is not the array of numbers but how that array changes when you switch coordinates: every upper index transforms with the Jacobian of the new-in-terms-of-old map and every lower index with the inverse Jacobian (the tensor transformation law). This rule is the membership test for tensorhood. The same object's components in different frames are different arrays, yet they all encode one frame-free thing, and any equation written as 'tensor = tensor' with matching index structure is automatically true in all frames at once.
Tensors are the right language whenever a physical quantity has directional structure beyond a single arrow: stress in a beam (force per area depends on two directions — the face and the force), the strain in a deformed solid, the moment of inertia of a rigid body, the electromagnetic field, and above all the metric and curvature of spacetime in general relativity. The whole appeal is that writing physics as tensor equations guarantees the laws look the same in every coordinate system — the principle of general covariance. The common beginner trap: a list of numbers with indices is NOT automatically a tensor; the Christoffel symbols, for instance, carry indices but do not transform tensorially, so they are not a tensor.
The stress tensor sigma^{ij} in a solid is rank 2: sigma^{ij} is the i-th component of the force per unit area acting on the face whose outward normal points in the j-th direction. Feed it the unit normal n_j of any surface and it returns the traction vector t^i = sigma^{ij} n_j — the actual force per area on that surface, for every orientation, from one object.
A rank-2 tensor is a machine that turns a direction into a vector — here a surface normal into its traction.
Possessing indices does not make something a tensor; the transformation law does. The connection coefficients (Christoffel symbols) carry indices but pick up an extra inhomogeneous term under coordinate change, so they are emphatically not tensors.