The other solutions posted already do a good job of explaining tensors as dyads, linear transformations, or using the ellipsoid interpretation. I thought I'd give an engineering perspective.
In engineering applications, physical quantities (mass, velocity, force, etc.) are either "scalars", "vectors", or "tensors". What determines which one, is how the quantity changes under a coordinate transformation. In other words: If you rotate your world view, what happens to the quantity in question? Suppose $U$ is a rotation (read: orthogonal) matrix that rotates vectors from the initial frame $F_1$ to another frame $F_2$.
Scalars, such as mass, do not change at all. An object's mass does not depend on the orientation of your frame, so $m_1$ = $m_2$.
Vectors, such as force or velocity, do change. If $v_1$ is expressed in your starting frame, then $v_2 = U^Tv_1$. Rotating the frame amounts to rotating the vector. All vector quantities tranform in exactly this way.
Tensors, such as stress or inertia, change as well. If $J_1$ is expressed in your starting frame, then $J_2 = U^TJ_1U$. Any quantity that transforms in this manner is a tensor.
Angular momentum (the product of the moment of inertia and the angular velocity) is a vector, because:
$$
J_2 w_2 = U^TJ_1U U^Tw_1 = U^T(J_1 w_1)
$$
Kinetic energy is a scalar, which is intuitive, but you can verify it using the transformations above. For an object in translational and rotational motion,
$$
E = \frac{1}{2} m_1 v_1^T v_1 + \frac{1}{2} w_1^T J_1 w_1 = \frac{1}{2} m_2 v_2^T v_2 + \frac{1}{2} w_2^T J_2 w_2
$$
And so on.