In physics we use the following (relevant to the tensor concept in three dimensional space, to make it simple):
scalars: these have a single value from the field of real numbers at each point is space . Example: temperature,
vectors: these have three values from the field of real numbers at each point in space and they obey vector algebra. They are useful to describe directional observations. example: force, velocity. (v_x,v_y,v_z) will describe the velocity and (v.v), the dot product of the vector is a single real number that is the speed. I suppose you are familiar with this.
In studying and measuring natural phenomena it was found that these were not enough in describing physical systems. Tensors were introduced, where for each point in space nine numbers are needed to describe the data. It is not enough to know the x component of the value under study, since it changes value ( at a fixed x) for different values the y axis and z axis.
All three obey particular coordinate transformation equations. Scalars do not change value, vectors and tensors follow the rules of the transformations.
A matrix representation for a tensor makes it simpler:
This example is a symmetric tensor , but good for an example. Looking at the columns it is like a vector, (x,y,z) components, looking at the row the same. It is used for physical quantities which differ in this "peculiar " manner and need all nine components to make sense. For example electric susceptibility in some crystal lattices.