What does this phrase mean? What is an equivalent expression in English, and could you give some examples of usage?
It can be used as an answer to "Grazie", like "prego", "non c'è di che", "figurati", but I dare say is the stronger one. To say something stronger you should come up with a long sentence, like
"Ma assolutamente, (non pensarci nemmeno) ci mancherebbe altro!".
Basically, you're refusing the idea the other person owes you something.
Its other meaning... well, the meaning is actually the same one, but the situation you're using it changes, so it doesn't appear so... Like I was saying, the "other meaning" is something like "that's out of discussion", "it's not gonna happen", "that's preposterous!". For example, you could say, "ci mancherebbe [altro] che dopo quello che è successo ti perdoni": "ci mancherebbe che ti perdoni" could be accurately translated as "there's no way I can forgive you".
The literal meaning is "(if what you're talking about would actually happen/go through/...) then something would be missing".
Imagine you've received an absurd request, and the other person tells you "Well, are you gonna do me this favor?". you could say "And then what? I could clean your apartment and call you master for the rest of my life!?!"
So, that's what's missing, an even more absurd request, which is unbelievable and still what you've been asked is not that different in its absurdity.
Finally, there's another use, the one Matteo was talking about in a comment. This too comes after a request, but you can use it to express your agreement (!).
Yeah, the very opposite of the previous one.
It depends on context, tone of voice etc. but you can also be more explicit.
The basic idea is the same one: "(If......) then something would be missing". Let's say someone asks you to stay at your place. You have two choice. You could answer
Ma ci mancherebbe altro (che ti ospitassi), non pensarci nemmeno!
The absurd idea is that you let him stay ("che ti ospitassi" means "that I let you stay". Here you're refusing the request, as we already said.
However you could answer with something apparently similar, that means the opposite
Ma ci mancherebbe altro (che NON ti ospitassi), non devi nemmeno chiederlo!
This time, the absurd idea you can express or not, as you like, the one between parentheses, is the opposite, "che NON ti ospitassi" (that I won't let you stay). So with your expression "ci mancherebbe altro" you're saying that the opposite of the request you received is an absurd idea!
Do remember you can always add something to be perfectly clear.
In this last example you can add "you don't even have to ask!" (non devi nemmeno chiederlo) to refuse the idea you may not listen to the other person's request.
Likewise you may add "don't even think about it" in the previous one, when your intention was to refuse the request.