Is it permissible for me to use our neighbors' wireless internet, after they gave permission to my brother and his friend to use it, or must I seek their permission personally? And if my action was wrong, what is its expiation?

It is not permissible to connect to another person's Wi-Fi network without their permission. The reference for knowing the permission and its limits is the owner of the network. If the permission is general for the household, then its use is permissible; otherwise, it is not. Whoever uses the network without permission must repent to Allah and seek absolution from the network owner.