In linear algebra, for a given linear transformation (a certain kind of matrix which generally represents some operation) M, an eigenvalue of M is a scalar c such that given a non-zero vector v (called the eigenvector), Mv = cv, where multiplication on the left is matrix multiplication and multiplication on the right is multiplication of v by the scalar c.
The definition is significant because it says that for a certain vector v, the transformation M, no matter how complicated it may be, just scales v by a factor of c. This is useful, for instance, if you want to determine an axis by which to evaluate the range of the transformation, because by choosing an eigenvector, you are choosing a "simple" or "natural" perspective from which to evaluate the range.
The definition is significant because it says that for a certain vector v, the transformation M, no matter how complicated it may be, just scales v by a factor of c. This is useful, for instance, if you want to determine an axis by which to evaluate the range of the transformation, because by choosing an eigenvector, you are choosing a "simple" or "natural" perspective from which to evaluate the range.