Since Wikipedia does already a good job explaining the definition of the spectrum, here is a short answer to (2):
Operators are complicated objects, so the basic strategy of introducing the spectrum of an operator is to associate a much simpler object, subsets of the complex plane, to a very complicated object, the operator, in such a way that it is possible to learn about properties of the operator by studying the spectrum.
In terms of differential and integral operators, the spectrum and its classification is about the question when a differential resp. integral equation has a solution, and what does go wrong if there is no solution.
In applications to physics, especially quantum physics, eigenvalues and their eigenvectors correspond to possible states of a given system.
On a more advanced level, symmetries of quantum physical systems are usually expressed by the invariance of physical states under transformations of representations of the symmetry group, where a representation of the symmetry group is a (strongly continuous) representation in the group of unital operators on a given Hilbert space. Properties of the physical system are encoded in properties of the spectral measure of the group representation, like, for example, the "mass gap" of the Yang-Mills millenium problem: In Minkowski spacetime the existence of a "mass gap" is equivalent with the property that the spectral measure of the translation group is bounded away from zero.