One of the other answers briefly mentions tensors, but I think it’s a perfect answer to the question and deserves expansion. Tensors can be defined in several ways that are equivalent, but one’s choice of definition strongly influences how one thinks of tensors.

Tensor products of vector spaces can be defined by:

1. Universal property

2. Span of $v_1 \otimes \dotsb \otimes v_n$s, with some multilinearity relations

3. Space of multidimensional arrays (i.e., with choices of ordered bases for the factor spaces)

4. Gadgets that eat vectors and/or covectors, and spit out some other vectors (or covectors or numbers)

5. Objects that transform in certain ways under changes of coordinates

The same things come up for tensor fields (or tensor bundles).

One of the other answers briefly mentions tensors, but I think it’s a perfect answer to the question and deserves expansion. Tensors can be defined in several ways that are equivalent, but one’s choice of definition strongly influences how one thinks of tensors.

Tensor products of vector spaces can be defined by:

1. Universal property,

2. Span of $(v_1 \otimes \dotsb \otimes v_n)$s, with some multilinearity relations,

3. Space of multidimensional arrays (i.e., with choices of ordered bases for the factor spaces), or

4. Gadgets that eat vectors and/or covectors, and spit out some other vectors (or covectors or numbers).

Similar definitions are available for tensor fields (or tensor products of bundles), and in that case there is additionally the "definition" of a tensor field as

5. An object that transforms in certain ways under changes of coordinates.