Once you know what a tensor is we can define a tensor field by varying it over a space in the same way as we think of a vector field.
There are two ways in which vectors are generally introduced in mathematics,
geometrically as orientated lines
and algebraically following the axioms of a vector space.
The physics of continuous materials and GR adds a third, its defined transformationally through its components. This relies on the local symmetry group of space.
Likewise, we can do the same for tensors - that is they have a geometric, algebraic and transformational definition. Here I'll concentrate on the geometric.
Vectors are usually introduced as a directed length.
A 2-tensor, geometrically speaking, is basically a directed area. They are defined by giving two vectors and these form the edges of the parallelogram, which we think of as exactly the 2-tensor. When generalised to higher spaces, Lang calls these blocks. So this parallelogram is a 2-block. We can see geometrically, that the 2-block depends linearly on the two edges. Thus we have 2-linearity.
Whereas a vector can scale in only one direction, a 2-tensor can scale in two independent directions; however, a 2-tensor can also internally scale. That is we can shrink one edge by a factor s, and expand the other edge by the same factor and this will give the same tensor. This phenomena is not seen in 1-tensors, aka vectors, since there is only one direction.
Now, although vectors are introduced as directed lengths, the length of a vector is actually additional information - this is the norm. Thus we ought to think of a vector as a directed scaling; and a 2-tensor as a 2-directed 2-scaling. Likewise for higher tensors.
In fact, there are two notions of length available for vectors. One is the unorientated notion of length, the norm, and this is the usual notion of length; and the second are the orientated notion of lengths. These are elements of the dual space and are usually known as covectors. Physically, they represent the potential energy of a particle moving along the vector in a uniform gravitational field.
For tensors, it isn't the unorientated length that generalises, that is the norm; but the orientated version. These are what are known as k-covectors (though really they ought to be called k-cotensors here since k-covectors are usually alternating). This geometrically speaking, gives the actual orientated area of an area element - it is a number, whilst the area element, like a vector is geometrical.
Categorically, a tensor is defined by a certain diagram and this simply says that any area function on the defining edges is the same as an area function on the tensor block.
A tensor field is then a field of tensors, that is a smoothly varying distribution of tensors over a manifold.
In terms of formalism, let M be a manifold
Then XM, the sheaf of vector fields, is the sheaf of sections of the tangent bundle TM over M, the base manifold. And it is a module over CM, the space of smooth functions on M. In fact, a sheaf of modules.
If we write $X^p_q$.M, for the sheaf of tensor fields of type$ (p,q)$, that is p-contravariant, and q-covariant, then this is the sheaf of sections of the $(p,q)$-tensor bundle $(x)^p_q.M := [(x)^p.TM] (x) [(x)^q.T*M].$ These are also module over CM.
It's probably useful to realise that the calculus of differential forms is basically the same as antisymmetric covariant tensor fields.