question on left and right eigenvectors

I know that $A\textbf{x}=\lambda \textbf{x}$, where $\textbf{x}$ is right eigenvector, while in $\textbf{y}A =\lambda \textbf{y}$, $\textbf{y}$ is left eigenvector.

But What is significance of left and right eigenvectors ? How do they differ from each other geometrically?

The (right) eigenvectors for $A$ correspond to lines through the origin that are sent to themselves (or $\{0\}$) under the action $x\mapsto Ax$. The action $y\mapsto yA$ for row vectors corresponds to an action of $A$ on hyperplanes: each row vector $y$ defines a hyperplane $H$ given by $H=\{\text{column vectors }x: yx=0\}$. The action $y\mapsto yA$ sends the hyperplane $H$ defined by $y$ to a hyperplane $H'$ given by $H'=\{x: Ax\in H\}$. (This is because $(yA)x=0$ iff $y(Ax)=0$.) A left eigenvector for $A$, then, corresponds to a hyperplane fixed by this action.

The set of left eigenvectors and right eigenvectors together form what is known as a Dual Basis and Basis pair.

http://en.wikipedia.org/wiki/Dual_basis

In simpler terms, if you arrange the right eigenvectors as columns of a matrix B, and arrange the left eigenvectors as rows of a matrix C, then BC = I, in other words B is the inverse of C

Geometrically the matrix $A$ is an origin and line preserving transformation (${\bf v}\mapsto A\cdot{\bf v}$). The right eigenvectors are eigenvectors for this transformation, but the left ones for $A^T$, which, geometrically can be totally different.

However, the eigenvalues and the dimensions of their corresponding eigenspaces must stay the same.