How can I partition a graph such that as few edges as possible cross partition boundaries?
Your problem is known as graph partition. Apart from $n$, the problem has a parameter $\epsilon \geq 0$, and it is required that each part have size at most $(1+\epsilon)|V|/n$. The Wikipedia article mentions several relevant algorithms. The case $n = 2$ is known as minimum bisection when the parts are required to be exactly equal, and this is a well-known special case.