The algebraic multiplicity of an eigenvalue is the number of times it appears as a root of the characteristic polynomial (i.e., the polynomial whose roots are the eigenvalues of a matrix).

Is eigenspace the same as eigenvector?

is that eigenspace is (linear algebra) a set of the eigenvectors associated with a particular eigenvalue, together with the zero vector while eigenvector is (linear algebra) a vector that is not rotated under a given linear transformation; a left or right eigenvector depending on context.

What do you mean by Eigen space?

An eigenspace is the collection of eigenvectors associated with each eigenvalue for the linear transformation applied to the eigenvector. The linear transformation is often a square matrix (a matrix that has the same number of columns as it does rows).