Here is the one I heard recently.
Professor: What is a root of $f(z)$ of multiplicity $k$?
Student: It is a number $a$ such that if you plug it into $f$, you get $0$; if you plug it in again, you again get $0$, and so $k$ times. But if you plug it into $f$ for the $k+1$-st time, you do not get $0$.
