Linear Algebra Eigenspace

I'm really confusing myself with eigenvalues, eigenvectors, and eigenspaces. So , say you have a 3x3 matrix and find the eigenvalues. One eigenvalue is 1 with a multiplicity of 1 and another eigenvalue is 2 with multiplicity 2. Do these different eigenvalues result in a different eigenspaces? I know the bases for each eigenvalue would be different, and from what I gather the basis for an eigenspace IS the basis for an eigenspace?? Also, does the multiplicity of an eigenvalue correspond to it's dimension/nullspace? Hopefully I'm making sense. Any help is much appreciated!