Means: A shrug of indifference
Usage: "Shouldn't you check the homework before you hand it in?" "CBA."
Clean up the abject tip that is your room? Discreetly dispose of the drink cans scattered across the playing field? "Nah. CBA".
Blah, bleh, deurrr, whatevah . It's come to this: surly, moody teens can't even be bothered to say they can't be bothered, so have abbreviated it to yet another TLA (that's "three-letter abbreviation", by the way).
Are they using the initials to play down the rudeness of their chosen expression, or is it - much more likely - that even "bovvered?" is too long-winded for them, too tedious to articulate. In the old days teachers used words like "insolence" and "laziness"; now it's called (de)motivation and (bad) attitude and we seem to have to put up with it.
The letters in full, of course, stand for "can't be arsed", though some users will tell you, either being coy or due to ignorance, that it's "can't be asked". Some even find CBA too long, abbreviating that to "ceebs". Now that really is CBA.