You know how to pronouce it, but what does it really mean?
It is actually a noun.
Originially, opposition to the disestablishment of the Church of England, now opposition to the belief that there should no longer be an official church in a country.
So what does 'disestablishment' mean?
Disestablish: dis·es·tab·lish tr.v.
1. To alter the status of (something established by authority or general acceptance).
2. To deprive (a church) of official governmental support.
And so Disestablishment (n) is simply the act of terminating an established state of affairs.
Ok, so to put it another way, antidisestablishmentarianism is the opposition to separation of church and state. Basically (this is what I get out of it anyways) it is wanting to have the church play a role in the government. I think that's what it's all about. There's a lot of double negatives in that word so I dunno.
Either way, you're now that much better of a person for knowing (somewhat at least) what that word means. But that also means you have to remember what the definition is. Don't forget to remember.