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?