Just as a natural transformation between functors is a collection of $1$-cells indexed by $0$-cells, a modification between transformations is an indexed collection of 2-cells.

Definitions

To be most general, start with lax transformations$\alpha, \beta : F \dot{\to} G : C \to D$. Given those, a modification $m : \alpha \ddot{\to} \beta$ assigns to each $0$-cell $A\in
C$ a $2$-cell $m_A : \alpha_A \Rightarrow \beta_A$ in $D$ that commutes suitably with the $2$-cell components of $\alpha$ and $\beta$ (see Leinster for details).

If $\alpha,\beta$ are strict transformations, then the complicated-looking modification condition becomes simply a naturality square with globs $m_A : \alpha_A \Rightarrow
\beta_A$ where the $1$-cells $\alpha_A$ would normally be.

Generalisation

When you get tired of thinking individually about $n$-categories, functors, transformations, modifications, and so on, check out (n,k)-transformation.

(Some discussion from here has also been moved to there.)