Why Tensors?
The principle of general covariance states that the laws of physics should take the same mathematical form in all reference frames and was one of the central principles in the development of general relativity. The term 'general covariance' was used in the early formulation of general relativity, but is now referred to by many as diffeomorphism covariance. Although diffeomorphism covariance is not the defining feature of general relativity, and controversies remain regarding its present status in GR, the invariance property of physical laws implied in the principle coupled with the fact that the theory is essentially geometrical in character (making use of geometries which are not Euclidean) suggested that general relativity be formulated using the language of tensors. This will be discussed further below.
Read more about this topic: Mathematics Of General Relativity