The algebraic multiplicity of an eigenvalue $\lambda$ is the number of times $\lambda$ appears as a root of the characteristic polynomial.
The geometric multiplicity of an eigenvalue $\lambda$ is dimension of the eigenspace of the eigenvalue $\lambda$.
Let us consider the linear transformation $T:\Bbb R^3 \to \Bbb R^3$ for simplicity. Suppose the characteristic polynomial of $T$ has the eigenvalue $\lambda$ as a repeated root, $2$ times. For example, if the eigenspace of the eigenvalue $\lambda$ were a line (one-dimensional), we could visualize $T$ as the transformation squishing or stretching all vectors on that line by an amount $\lambda$. But what is the geometric significance of the algebraic multiplicity $2$, in this case? Is there any underlying geometric intuition?
As you mentioned, the geometric multiplicity of an eigenvalue $\lambda$ is the dimension of its eigenspace. The algebraic multiplicity is the dimension of what's called the generalized eigenspace.
If you have a linear transformation, represented by a matrix A, and an eigenvalue $\lambda$, a generalized eigenvector is a vector, $v$ such that for some integer n, $$(A-\lambda I)^nv =0$$
Then, the space formed by taking all such generalized eigenvectors is called the generalized eigenspace and its dimension is the algebraic multiplicity of $\lambda$.
There's a nice discussion of the intuition behind generalized eigenvectors here.