It really has two meanings, both of which make no sense, but judging by where you've placed the apostrophes, it's basically saying tomorrow and yesterday.
'Tomorrow's Today' means the 'today' of tomorrow, which would, in fact, be tomorrow. I don't know how to explain it any more clearly. 'Today' means the present day. So 'Tomorrow's Today' is the present day, but tomorrow. I know, that really doesn't make any sense, but you'll get it eventually.
Today's Yesterday is saying almost the same thing, but in reverse. 'Today's Yesterday' is yesterday; it's the yesterday of today.
So, as I've said so many times already, I know this barely made any sense, but yeah... it's the best I could explain it.