I think it was Gregorio Ricci-Curbastro. His absolute differential calculus became the foundation of tensor analysis. Please see the link.
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Ricci-Curbastro.html
Romer is right: Ricci-Curbastro was the founder of Tensor Analysis (he called it "absolute differential calculus"), but, as Einstein said in the introduction of his 1916 GR paper, the methods of Ricci were improved into a systematic framework by Tullio Levi-Civita(cfr. http://einsteinpapers.press.princeton.edu/vol6-trans/158).
The seminal paper on tensors is "Méthodes de calcul differéntiel absolu et leurs applications" (Math Ann. 54, 1901) a joint work by Ricci and Levi-Civita.
The latter defined also the concept of parallel displacement in curve spaces.
The idea of tensors was born from of study of curvature. This was done by B. Riemann in A mathematical word that seeks to answer the question posed by the most distinguished Academy of Paris.
Just I know that it was developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his theory of general relativity.
For a complete answer, please refer to the following book of mine:
Tensors:Mathematics of Differential Geometry and Relativity. Prentice Hall of India Learning Pvt. Ltd., New Delhi (may 2015)
The complete answer of the question about the Tensors is as follows:
In many areas of mathematical, physical and engineering sciences, it is often necessary to consider two types of quantities. First, those which have magnitude only. Such quantities are known as scalars. Mass, length, volume, density, work, electric charge, time, temperature, etc. are the examples of scalars. The second type are those which have both magnitude and direction. These are known as vectors. Some of the examples of vectors are velocity, acceleration, force, momentum, etc.
Quite often the notion of vector is not sufficient to represent a physical quantity. What happens when we need to keep the track of two (or more) pieces of information for a given physical quantity? For such situation, we need a tensor. A tensor contains the information about the directions and the magnitudes in those directions. Thus, for example, the stress at a point depends upon two directions; one normal to the surface and the other that represents the force creating stress and thus stress cannot be described by a vector quantity. As another example, the measurement of charge density will depend upon the four velocity of the observer and thus can be represented by a vector, while the measurement of electric field strength in some direction will not only depend upon this direction but also on the four velocity of the observer and thus such measurement cannot be described by a vector quantity alone. These and similar other examples led to the generalization of a vector quantity to a quantity known as tensor.
Life would have been miserable without tensors as we can not walk across a room without using a tensor (the pressure tensor), it is impossible to align the wheels on our car without using a tensor (the inertia tensor) and definitely one can not understand Einstein's theory of gravity without using tensors. The word ``tensor" itself was introduced in 1846 by William Rowan Hamilton to describe something different from what is now meant by a tensor. The contemporary usage was brought in by Woldemar Voigt in 1898. The word "tensor" is derived from the Latin word "tensus" meaning stress} or tension. In anatomy the word tensor means a muscle that stretches or tightens some part of the body.
The concept of tensors has its origin in the development of differential geometry by Gauss, Riemann and Christoffel. The tensor calculus (also known as absolute calculus) was developed around 1890 by Gregorio Ricci-Curbastro and originally presented by Ricci in 1892. It was made accessible to many mathematicians by the publication of Ricci and his pupil Tullio Levi-Civita's classic text "Methodes de calcul differentiel absolu et leurs applications" (Methods of absolute differential calculus and their applications) [Mathematische Annalen Vol. 54 (1901)]. In the 20th century, the subject came to be known as tensor analysis, and became popular when Albert Einstein used it in his general theory of relativity around 1915. General relativity is formulated completely in the language of tensors. In Einstein's general relativity, using Riemannian geometry, gravity-inhabited space is analysed. The mass manifests as curvature in the spacetime geometry. Riemannian geometry (geometry of the curved space), until that time, was thought to be an abstract mathematical structure with no physical use. With the tools of Riemannian geometry, Einstein was able to formulate a theory that predicts the behaviour of objects in the presence of gravity, electromagnetic and other forces.
Einstein had learned about tensors, with great difficulty, from the geometer Marcel Grossmann. Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis.
Tensors were also found to be useful in other fields such as continuum mechanics. Some well-known examples of tensors in differential geometry are quadratic forms such as metric tensors, and the Riemann curvature tensor. The exterior algebra of Hermann Grassmann, from the middle of the nineteenth century, is itself a tensor theory and highly geometric; but it was some time before it was seen, with the theory of differential forms, as naturally unified with tensor calculus. The work of Elie Cartan made differential forms one of the basic kinds of tensors used in mathematics. From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology. There are different types of tensor concepts that frequently occur in many branches of abstract algebra, particularly in homological algebra and representation theory. Multilinear algebra can be developed in greater generality than for scalars coming from a field, but the theory is then certainly less geometric, and computations are more technical and less algorithmic. Tensors are generalized within category theory by means of the concept of monoidal category, from the 1960s.
Nowadays tensors have many applications in most of the branches of theoretical physics and engineering, such as mechanics, fluid mechanics, elasticity, plasticity and electromagnetism etc.
For more details about tensors, please refer to the following book of mine:
Tensors : Mathematics of Differential Geometry and Relativity. Prentice Hall of India Learning Pvt. Ltd., New Delhi (May 2015).
This must be Gregorio Ricci-Cubastro. Actually, Tensor calculus was initially referred to as Ricci-Calculus.
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