What is an ergative language? What is the ergative case? What is the absolutive case? What is ergativity?
First and foremost a warning: this is a very deep topic and an FAQ answer will not be able to provide a full picture.
To understand ergativity, you first must understand syntactic roles, and for that in turn you must know what transitive and intransitive verbs are. Luckily, this isn’t very difficult. Consider the following two sentences:
John slept.
Alice hit Bob.
The first is an intransitive sentence. It has merely one mandatory noun phrase, here “John”, which we shall label S. You can think of this as “subject”, but be aware that said word has a somewhat different meaning. The second sentence meanwhile has two noun phrases: “Alice”, who is doing the action, and “Bob” who is undergoing the action. We’ll give these roles names too: A (like actor) and P (like patient). Again however A does not strictly mean “agent”!
Those are the syntactic roles then: S for the sole argument (mandatory noun phrase) of an intransitive verb, and A, P for the two arguments of a transitive verb.
Now, if you speak an european language, you may find it odd that we distinguish between S and A. After all in English/German/Latin grammar these are considered the same, right? But that’s exactly where ergativity comes in. The short answer here is that ergativity is the name of any system wherein the A is treated one way (e.g. being marked by the ergative case) and S, P are treated another way (e.g. being marked by the absolutive case). The opposite of ergativity is accusativity, in which S, A are treated the same (marked with nominative) and P is given special treatment (marked with the accusative). I shall now go into a bit more detail.
Why ergativity? What is split-ergativity? What is tripartite alignment?
(read the answer to the previous question first) Consider a language where S, A, P are all marked with different cases. It might look something like this:
John slept.
Alice-a hit Bob-o
where -a marks the ergative case and -o marks the accusative case. Such a system is called tripartite, and while pretty liked among conlangers, tripartite systems are very rare in natural languages. Why might that be? The main reason is that the case-marking A and P are redundant. If you already know Alice is the actor, then there’s no need to highlight Bob as the patient. And vice-versa, if you’ve marked Bob as the patient, then you don’t need to highlight Alice as well. I’ve already chosen not to put a case marker on John at all, and this follows the common pattern in natural languages: because it does not have to be contrasted with anything, whichever case marks the S role will usually be the least marked, often not marked at all.
So, the sensible approach then is to pick either A or P as worth highlighting, and drop the marking on the other. Highlighting A gives ergativity, highlighting P gives accusativity. But are there any reasons to choose one over the other? The answer is of course yes. In general, noun phrases referring to very animate objects (such as personal pronouns or nouns describing humans) will most commonly be in active roles in a sentence. Thus, their expected role in transitive sentences is A. As such, for these nouns, it makes sense to highlight it whenever they are not in this active role, i.e. use accusative marking. Meanwhile, very inanimate nouns like “rock” or “tree” will more often be the affected object in a sentence, as they don’t typically initiate actions. As such for them the expected role is P, and ergative marking is the sensible choice to highlight the exceptions to the rule.
So you may wonder if you have to choose now. And the answer is a resounding no. In fact, ergativity and accusativity are the extrema on a spectrum called split-ergativity. Very few languages are thoroughly ergative, and the spectrum is heavily skewed towards the accusative end, but many languages are somewhere in the middle. One common “split” is based on the animacy of the involved noun phrases, as I described above. In such a system, it may be that animate nouns and pronouns receive accusative marking, while inanimate nouns receive ergative markings. There may even be a class of nouns in the middle where the two systems overlap and you get a bit of tripartite marking, which is always fun.
Another possibility for splitting is based on the verb involved. Here, for transitive verbs there are simply two cases marked (possibly one moreso than the other), but for intransitive verbs more active participants will be marked with the same case as A (thus giving an accusative system), while more passive participants will be marked like P, giving ergativity. Which case is chosen can depend either soley on the verb (termed Split-S) or on the specific circumstances such as the amount of control the noun has (termed Fluid-S), or on both.
Further kinds of splits involve having present or imperfective sentences be accusative and past or perfective be ergative; or to use different systems in main and subordinate or relative clauses. These things can also all be combined, of course.
The last thing to note is that so far we’ve only talked about morphological ergativity. On top of that there is also syntactic ergativity. For this, I shall simply refer to this amazing write up on syntactic alignment by /u/Gufferdk: Dive Deeper - Syntactic Alignment and Pivot Constraints
Follow-Up: What is the difference between S/A/P and Subject/Agent/Patient?
S/A/P refer to syntactic roles. That is, it does not actually matter what kind of noun fills them or what role it really plays in the semantic interpretation of the sentence.
Subject is simply a term for something that is shared by the roles of S and A.
Agent and Patient are semantic roles: they care about the actual meaning of the sentence. The discrepancy between semantic and syntactic roles is probably best shown with a passive verb. Passive verbs are intranstive verbs formed from a transitive verb by removing the A and moving P to S.
In Alice hit Bob, Alice is both A and Agent, Bob is both P and Patient. But forming the passive: Bob was hit by Alice, the semantic roles (agent, patient) remain unchanged, but Bob is now the S, and Alice is a so-called non-core or peripheral argument.