nLab
modification

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.)

References

Tom Leinster, Basic bicategories, arXiv.