Please help me understand how this works:

Julia Brown has a Disrupt that allows the player to sacrifice her and replace her with another Julia Brown or Sleepwalker if she loses a Terror struggle.

This takes place but my opponent plays Power Drain that cancels a response or action, and my opponent plays it to prevent Julia from swapping.

The rules say that a Disrupt is a special kind of Response, so I thought the Power Drain should stop Julia. Is that correct? Does a Disrupt disrupt and Disrupt?

Power Drain cannot target a Disrupt. There are four triggered effects called Action, Response, Forced Response and Disrupt (see page 13 of the LCG rules). Power Drain only targets Actions and Responses.

For contrast, two CCG cards Performance Artist and Court of Yhtill could have cancelled Julia Brown's Disrupt because they target a character's abilities or any triggered effect, respectively.

So, in answer to your question, the rules definitely allow a player to Disrupt a Disrupt effect, but Power Drain isn't the card to do it with.

cards that can disrupt a dirupt -

aldebaran ascendant

wrack

writhing wall

plus performance artist and court of yhtill.