eigenspace
A single eigenvector is one direction that A merely stretches. But if v is an eigenvector for lambda, so is 2v, so is -v, and so is any combination of independent eigenvectors that share that lambda. Collect all of them — every vector that A scales by the same lambda, plus the zero vector to make it a subspace — and you get the eigenspace of lambda.
Precisely, the eigenspace E_lambda = ker(A - lambda I) = { v : A v = lambda v }. It is a genuine subspace: closed under addition and scalar multiplication, and it always contains 0. Its dimension is exactly the geometric multiplicity of lambda. The nonzero vectors in E_lambda are precisely the eigenvectors for lambda.
Eigenspaces are the building blocks of diagonalization. Eigenspaces for distinct eigenvalues are independent (they intersect only in 0), so a basis of eigenvectors is built by stacking bases of each eigenspace. When those bases together span the whole space, A is diagonalizable and the space decomposes as a direct sum of eigenspaces.
The eigenspace is just the null space of A - lambda I; solve that homogeneous system to find it.
The zero vector lives in every eigenspace but is never called an eigenvector — by convention eigenvectors must be nonzero, otherwise 'A 0 = lambda 0' would hold for every lambda.