In a more general setting, let O be an operator, acting on a space T (usually a topological vector space, but anything with a scalar product will do). We say a scalar λ is an eigenvalue for O if there exists ψ ∈ T, such that Oψ = λψ. Here ψ is an eigen(vector/function/whatever you're calling elements of T) for O.
Where a determinant and trace are well-defined, it can be shown that det O is the product of the eigenvalues and tr O is the sum. Even more usefully, the characteristic function, cO(λ) = det O - λI, where I is the identity operator, has roots (only) at the eigenvalues.
Over function spaces, eigenvalues become very important in quantum mechanics where they represent values of observables. For example, the (time-independent) Schrödinger equation can be represented as Hψ = Eψ where ψ is a wavefunction, and H is the Hamiltonian operator. Here, the eigenvalue, E, represents the energy of the system.