>>11016668A field is "some object" attached to every point in a space. This "some object" can be a scalar (like temperature), or a vector (like an electric field), or something more mathy like an inner product: at each point you can have a different rule for measuring lengths and angles of vectors starting at that point.
When you choose a coordinate system, the object can be described by a collection of numbers: a vector is an n-tuple of numbers usually written in a column, an inner product is given by a matrix, which is an nxn-tuple of numbers organized into a grid etc. Since we have this object at each point of the space, these numbers depend on the point and the field is actually a collection of functions, not just numbers. You know this from multivariable calculus: a vector field is essentially just a pair or a 3-tuple of functions, only you decided to understand them as coordinates of a vector sticked to a point.
When you change the coordinate system, this collection of functions will change completely. But since they describe the same object, only expressed differently, there must be some definite relation between the old collection and the new collection. A tensor field means that this relation is in some sense the simplest possible: the new functions are just linear combinations of the old functions, and the coefficients are derivatives of the coordinate change (i.e. entries in the Jacobian matrix).
tldr: At a fixed coordinate system geometric and physical objects are described by a number of functions. When coordinate system changes, these functions also change. When this change is the nicest possible (linear), the object is called a tensor (field).
