You know, I'm not sure this answer is all that helpful, and some of it seems a little disingenuous.
It is important to explain to someone the nature of nested hierarchies, and that descendants of verterbrates will always be a part of Vertebrata (at least, it is if that someone is actually interested in learning, otherwise it might just be a waste of time). However, I think we have to be clear about what we mean by this.
To be a member of Veterbrata simply means to be a descendant of an organism arbitrarily defined as the last common ancestor of Vertebrata, But there's a bit of equivocation here with the traditional meaning of vertebrate, which predates evolutionary classifcations, that of 'an animal with a backbone'. If, in the distant future, there's eome species knocking about that descends from verterbrates, but which does not possess a backbone, it would still be a member of Vertebrata, still a vertebrate phylogenetically, but it wouldn't fit the original defining concept of vertebrate any more. This is the sort of thing creationists are looking for when they protest 'but it's still a moth' (though of course they wouldn't define the key transition explicitly, and they seem to expect it to happen in an afternoon).
While it might be difficult to imagine vertebrates evolving into something without a backbone, for an example of the kind of thing I mean, look at the tunicates below. These aren't vertebrates, but they are chordates, and so closely related. Chordata is those animals with a notochord - a cartilaginous rod which, in vertebrates, is replaced by the vertebrae. Tunicates have a notochord in their larval form, but metamorphose into an adult form which doesn't have one, and in fact looks nothing like we'd expect from a close relative of vertebrates. Were a tunicate to evolve that did away with it's larval stage and skipped straight to the adult, repodcutive form, it would still be a chordate phylogenetically, but it would no longer look anything like most chordates and would not fit any description you usually see of what a chordate is, except one which only used phylogenetic criteria.
The names we have for types of organisms are based on the organisms we see around us at present. Saying the descendants of moths will always be moths is not really true. They'll always be Heterocera, the moth clade, but if they evolve into a form that doesn't look like a moth they're not really moths any more. Similarly, cows aren't fish, even though they're nested within them phylogenetically.
A less fundamental criticism - shrimp do not have a ventral notochord, they have no notochord at all. Only chordates have notochords, that's where the term 'chordate' comes from.
Edited by caffeine, : Just wanted to point out that the picture is taken from wikipedia's page on tunicates.
For example, if a invertebrate evolved a backbone would it be classified as a vertebrate?
You can solve this problem by defining 'vertebrae' as the structure that makes up the backbone of vertebrates, so that even if another animal is found with a structurally and functionally identical organ, it's bits wouldn't be vertebrae since it's not homologous with the organ in vertebrates.
This is the type of thinking involved when people argue that we shouldn't call a male intromittent organ a penis in a bird, a reptile, or a spider, since it's not homologous with the mammalian penis. Or that flying cats have AIDOs instead of wings.
Hence the problem with colloquial naming traditions. We arbitrarily decide what is an isn't a moth based on our feeling at the time.
Is a chihuahua a wolf? Most would say no. Did chiahuahuas evolve from a wolf-like ancestor? Most likely. We will say that chihuahuas and wolves are still dogs, but when did those wolves stop being wolves and become chihuahuas? That dividing line would also be an arbitrary one.
The system that science has settled on is cladistics, and in that system you never evolve out of your clade. This differs from the colloquial method where species are arbitrarily assigned to different groups based on shifting notions of what is or isn't indicative of a group.
Yes, that's one of the reasons we have terms like Heterocera when it comes to scientific nomenclature, instead of simply making do with 'moth', so that we can have a learly defined phylogenetic concept independent of people's arbitrary impressions of what a thing should be classified as.
But I don't think using this sort of argument helps when someone is protesting that a sloth is never seen to turn into a turtle. Saying that 'the descendants of sloths will always be sloths' isn't true - it's equivocating between a common name which we wouldn't apply to something that looks substantially different to a sloth, and the phylogenetic concept 'Folivora'.
amp1022's problem seems to be that he thinks the transition from fish to ostrich is supposed to happen in an afternoon and only involve one animal. Trying to explain phylogenetic nomenclature is more like to confuse than to help, at this point.