It should also be noted that many mathematical structures informally called 'tensors' are actually 'tensor fields', fields defined over a manifold which define a tensor at every point of the manifold. See the tensor article for an elementary introduction to tensors.
The geometric intuition for a vector field is of an 'arrow' attached to each point of a region, with variable length and direction. Our idea of a vector field on some curved space is supported by the example of a weather map showing horizontal wind velocity, at each point of the Earth's surface.
The general idea of tensor field combines the requirement of richer geometry  for example an ellipse varying from point to point  with the idea that we don't want our notion to depend on the particular method of mapping the surface. It should exist independently of latitude and longitude, or whatever particular 'cartographic projection' we are using to introduce numerical coordinates.
The contemporary mathematical expression of the idea of tensor field breaks it down into a twostep concept.
There is the idea of vector bundle[?], which is a natural idea of 'vector space depending on parameters'  the parameters being in a manifold. For example a vector space of one dimension depending on an angle could look like a Moebius band as well as a cylinder. Given a vector bundle V over M, the corresponding field concept is called a section of the bundle: for m varying over M, a choice of vector v_{m} in V_{m}, the vector space 'at' m.
Since the tensor product concept is independent of any choice of basis, taking the tensor product of two vector bundles on M is routine. Starting with the tangent bundle  the bundle of tangent spaces  the whole apparatus explained at componentfree treatment of tensors carries over in a routine way  again independently of coordinates, as mentioned in the introduction.
In the end, we can give a definition of tensor field, namely as a section of some tensor bundle. This is then guaranteed geometric content, since everything has been done in an intrinsic way.
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