Sentence:  The <WSD>tensor</WSD> \alpha_{ij} should really be called a “<WSD>tensor</WSD> of second rank,” because it has two indexes. A vector—with "one" index—is a <WSD>tensor</WSD> of the first rank, and a scalar—with no index—is a <WSD>tensor</WSD> of zero rank. 
Meaning of the word marked with <WSD> tags:  A mathematical object that describes linear relations on scalars, vectors, matrices and other tensors, and is represented as a multidimensional array.<ref>Rowland, Todd and Weisstein, Eric W., [http://mathworld.wolfram.com/Tensor.html "Tensor"], Wolfram MathWorld.</ref>