If you believe in schemes, take any smooth curve $C\subset \mathbb P^2$ in the projective plane and intersect it with its tangent line $T_P \subset \mathbb P^2$ at some arbitrary point $P\in C$.
The intersection $T_P\cap C$ will not be smooth at $P$ , since it is not reduced there.
If you only want to hear about good old varieties, consider the smooth surface $S\subset \mathbb P^3$ defined by the equation $ZT-XY=0$ and intersect it with the (obviously smooth) plane $P\subset \mathbb P^3$ given by the equation $Z=0$.
The intersection $I=P\cap S $ is the variety given by the two equations $Z=0$ and $XY=0$.
It consists of the union of the two lines $Z=X=0$ and $Z=Y=0$ .
This intersection variety $I$ is singular at the intersection $[0:0:0:1]$ of the two lines.