So this is my attempt to explain what Ā movement is. I hope it makes sense. And that there aren't too many mistakes
First pass
First thing: "Ā" is meant to be read "A-bar," and that's how I'll write it. (It's sometimes written "A'" instead.) And A-bar movement contrasts with simple A movement, the bar (or line above, or prime mark) indicating negation.
Second thing: the A/A-bar distinction is one of the worst naming conventions in all linguistics. Especially when you realise it has nothing to do with X-bar theory. It can also be maddening to try googling it.
The A stands for Argument, and the issue here is a contrast between movement that's somehow specific to a verb's arguments (A movement), and certain other sorts of movement (A-bar movement). The idea is that in many languages, there are syntactic requirements that one or more of a verb's arguments have to move to a particular position in the clause's structure, and that sort of movement is called A movement.
English has a requirement of this sort: a finite clause must have a syntactic subject, that is, an NP that occupies a particular position in the clause's structure, and most often this requirement is satisfied by moving one of a verb's arguments into that position. (There are also expletive subjects, which I won't talk about here.)
English also has a requirement that question words in non-echo non-quiz-show questions get moved to the top of the clause. This is
not A movement: it doesn't have to target arguments, it doesn't even have to target NPs, and its target is distinguished not by its position in the structure but by some additional characteristic or feature. That's to say, this is A-bar movement rather than A movement.
Second pass
I'm assuming a picture where a clause is assembled bit-by-bit, in such a way that the result is a tree structure (a binary tree structure, to be a bit more precise).
In this picture, a verb generally gets combined with its arguments in a fixed order. There's some room for variation (both within and between languages), but, for example, if a verb's core arguments include an agent and a patient, the verb will combine with the patient before the agent. As a result, the agent will start out higher in the tree structure than the patient does. (I won't try to include a diagram, but I hope it's easy to imagine the sort of tree structure I'm imagining. I'm also assuming that in a passive, even if there's a by-phrase or some equivalent, the overt agent doesn't count as a core argument of the verb.)
A verb's arguments need not stay in their initial ("merge") position, it's common for them to have to move higher in the clause as various other things get added. Some of these movements, the A movements, seem to have only a syntactic motivation. Like, English requires the subject to move, but Italian doesn't, but this by itself doesn't result in any semantic or pragmatic differences between English and Italian sentences, it seems to be pure syntax. And similarly in other languages that require object movement.
When I say that it seems to be pure syntax, in part I mean that no one has any really good idea why it happens. Like, the English requirement that subjects occupy a particular position is sometimes called the extended projection principle (EPP), but that's just a label (and a terrible one), not an explanation. And the same is true for other terminology that's used to describe this sort of phenomenon---like abstract case, or strong features, or nominal licensing.
Further, it seems to be purely a syntactic question which NP gets moved by A movement. For example, when English moves an argument into subject position, it'll always move the agent if there is one, but if there's only a patient, then it'll move the patient. That is, it's sensitive to the position of the argument in the sentence's structure (agents are merged higher than patients), not to semantic role. (But in a language where objects also A move, it can be tricky to explain how this happens the way it does; and that's not the only sort of puzzle that can arise.)
In any case, A movement is the movement of one of a verb's core arguments from its original merge position to some higher position in the clause, and it happens for purely syntactic reasons, whatever exactly that means. Whereas A-bar movement need not involve a verb's core arguments, and generally happens because the thing that move has some particular, often pragmatic, significance.
Kinds of A-bar movement
There are various common sorts of A-bar movement, which I'll run through now.
I've already mentioned question words, and in languages that allow fronted question words that can be one of the plainest sorts of A-bar movement. Here's an example:
(1) Whatᵢ did I eat tᵢ?
(I'm using the common convention of using
t to show the earlier position of a moved element, and a subscript to make it clear which overt element it corresponds to.)
The idea is that in this sentence, "what" has moved from after (lower than) the verb to a position high in the clause (the front). This obviously doesn't satisfy a general syntactic requirement on objects. Instead, it happens because "what" is a question word, and in a non-echo, non-quiz-show question, English requires question words to move to the top.
In many languages, including English, you also get A-bar movement in relative clauses. That's to say, there's a gap in the relative clause from which an operator of some sort moves to the front of the clause. When there's an overt relative pronoun, this is usually taken to be the element that's moved:
(2) the person [who₁ I saw t₁]
But it's also possible for the moved element not to be pronounced:
(3) the person [∅ᵢ I saw t₁]
(Why think that something has moved in that case? One reason is that you get island effects regardless of whether an overt relative pronoun is present: "the person who₁|∅₁ the claim that I saw
t₁ surprised her" is bad either with or without "who." Another sign that there's movement here is that you can get parasitic gaps: "the person ∅́∅₁ I saw
t₁ without greeting
t₁." ---I'll come back to parasitic gaps, but not to islands.)
You also get A-bar movement in some forms of topicalisation. My English allows this, at least for contrastive topics:
(4) The apple I ate (but not the banana)
However, many languages allow sorts of topicalisation that probably don't involve movement at all. English "as for" topics are of this sort, for example:
(5) As for the apple, I ate it
(Note that you need the resumptive pronoun here, and there'll often be a pause where I've put a comma.)
Finally, I'll mention focus movement. I don't think we do this in English, but it's pretty common for there to be a position in a clause to which focused elements will move. That can be the front of the clause, but in SOV languages it's also common to have a focus position immediately before the verb.
Diagnostics
I'm going to mention a couple of diagnostics that can be used to distinguish A movement from A-bar movement. I find this stuff pretty hard to get my head around, and am a bit nervous that I'll screw up, but here goes.
First, weak crossover.
Take this example:
(6) ?Her₁ friends admire Mary₁
This is supposed to be a bad, maybe unacceptable, on the assumption it's saying that
Mary's friends admire Mary. (It's totally fine if it's supposed to be talking about someone else's friends.)
(7) ?Her₁ friends admire who₁?
And this is also supposed to be bad, on the assumption that it's asking for someone₁ whose friends admire her₁, the very same person. (You also have to interpret it as an echo question.)
(8) ?Who₁ do her₁ friends admire?
Again this is supposed to be bad, on the assumption that it's asking about someone who's admired by her own friends.
But this is fine:
(9) Who₁ is admired by her₁ friends?
(8) puts "who" before "her" by wh-movement, a sort of A-bar movement, and this isn't enough to let "who" bind (be coreferential with) "her." (9) also puts "who" before "her," but does so by passivisation: even before wh-movement (and even if this is an echo question with no overt wh-movement), "who" is the subject and will end up before "her" by plain A movement.
The fact that (8) is bad even though "who" has moved before "her" is an example of what's called weak crossover. Basically, this happens only with A-bar movement.
I'll also mention parasitic gaps.
Consider these two sentences:
(:) *I ate the apple without tasting.
(11) What did I eat without tasting?
In (:), but not in (11), you need "tasting" to have an overt object ("I ate it without tasting it" would be fine). In (11), the object is gapped, and this appears to be possible because the question word has left its own gap:
(12) What₁ did I eat t₁ without tasting t₁
The second gap here is called a parasitic gap, because it's supposed to be parasitic on the gap left by the movement of the question word.
And you can do this with other forms of A-bar movement:
(13) the apple that I ate without tasting
(14) The apple I ate without tasting
((13) illustrates a relative clause, (14) topicalisation.)
So there seems to be a general pattern, that A-bar movement does and A movement does not license parasitic gaps.
Final thoughts
There are a lot of loose ends there, but also this is already really long, so I won't try to address them in this post. But I do want to make one connection with ergativity, since bradrn was wondering about all this after reading about ergativity.
Many languages with ergative morphosyntax have restrictions of the following sort: with a transitive verb, the object can be A-bar moved, but the subject cannot. For example, you can't directly question the subject, or you can't focus it or relativise it. In fact this is the form that syntatic ergativity takes: usually it's specifically a restriction on A-bar movement. (You'll also see the movements in question called "extraction.") Syntactic ergativity doesn't always target all forms of A-bar movement, but it's most often one or more forms of A-bar movement that's at issue---so it's helpful to understand that in some ways these types of movement behave the same.
Incidentally, it's a big puzzle why so many morphosyntactically ergative languages have restrictions on A-bar movement. There are some analyses of some nonergative languages where there are analogous restrictions on the A-bar movement of objects, but this seems to be a rare phenomenon. (There's the Austronesian pattern that allows only one privileged argument to be extracted, and that argument is often identified as a subject, but this doesn't seem to be an accusative counterpart of syntactic ergativity.)
I'll mention another issue. It's not always clear which if any of a verb's arguments is the syntactic subject, in the sense that it obligatorily A-moves to a position high in the clause. Dyirbal, for example, often has patient-agent-verb order in clauses with two overt NP arguments, and you might think that the reason for its syntactic ergativity is that for some reason it's the patient argument that A moves into the subject position; and this sort of analysis has been given for a number of syntactically ergative languages.
In principle, one way to test this would be to try to apply diagnostics of the sort I've mentioned to see whether different word orders pattern differently with respect to scope or parasitic gaps or whatever. That's not likely to be easy, because grammars rarely provide the data that you'd need to settle questions like this, and in many languages it can be very hard to apply the tests. Like, one sort of test I haven't mentioned is the behaviour of words meaning
every; but most langugaes don't have a word meaning
every (they have
all instead). Or some languages let you drop topical objects in any context, making it impossible to test for parasitic gaps.
In fact there are other issues involving subjects and topics and A movement and A bar movement that have nothing to do with ergativity. But that's enough for now.