What's meant by: "All observers agree on the combination of basis vectors and components for a tensor"? There's a Youtube video given by Dr Dan Fleisch on Tensors, where he states at 11:22:

You may be wondering, what is it about the combination of components and basis vectors that makes tensors so powerful? The answer is this: All observers in all reference frames agree, not on the basis vectors, not on the components, but upon the combination of components and basis vectors"

What does he mean? Is he saying that as you change the combination of basis vectors, you also change the corresponding components?
 A: Yes. That's how the tensor calculus is formulated. For two observers on different frame of reference, all they have to see is the same physical laws. On this basis we formulate our physical laws so that it will be valid for any observers irrespective of their reference frames. Let's take a tensor of rank one (a vector) as example. Suppose two observers on frames A and B wants to measure the the distance vector between the points a and b:  
 
You could see that the individual position vectors each observer make with the points a and b is not the same as one could easily infer from the diagram. But both will observe the same distance vector between the points a and b. So even though the position vector is not a tensor (as it depends on the co-ordinate you are selecting since it is measured from the origin) the difference between two position vectors or say, the displacement vector is independent of the co-ordinate. So the basis have no crucial role in tensor calculus. tensors are a set of quantities whose components are independent of the co-ordinates. So the basis is allowed to change and at the same time the magnitude attached to the individual basis are also allowed to change. But overall, the vector should remain (or the tensor) unchanged with respect to different co-ordinates. Here we are speaking about co-ordinate symmetry. This came as a consequence because there is no standard reference frame for any scenario but the natural laws should be seen identical to every observers (otherwise saying the individual realities should coincide that's all matters, not how the just get into the same reality).
A: Yes; he is saying that a tensor encodes geometric information. A vector is a special case, and it's direction and legth are unchanged under a change of basis, though the components do change.
A more dramatic example is the determinant,  which gives the volume of the paralellopiped whose edges are defined by three coterminous vectors. A change of basis alters the components of the vectors, but not their lengths or angles -- the geometry remains unchanged. The determinant, which appears when written as a mess of components, is an invariant, and always gives the volume.
