Why do acids and bases both conduct electricity?
Because both of them have ions!
To conduct electricity, we require charged particles. In metals, free electrons are the charged particles which make it possible for electricity to be conducted.
Similarly, for acids and bases, there are ions. Note that they will only conduct electricity in their aqueous state, since the ions are mobile in this state.
Often, acids produce the
Similarly, bases often produce the
Have a look at this answer, which details how acids are able to conduct electricity.