Ergativity involves a difference between the
subjects of two kinds of sentence. Although English does not exhibit ergativity, I will illustrate it semantically using English in order to simplify things as grossly as possible. Take two sentences:
I work hard.
I work him hard.
In both examples, the word "I" is the subject of the sentence, and is in the "nominative" case.. In the first example, the verb "work" is intransitive - it takes no object. In the second example, the verb "work" is transitive - it takes the object "him", which is in the "accusative" case (the nominative case is "he", of course).
That is, "I work hard" means that working hard happens to me, but there is no other object to which "work" happens. But "I work him hard" means that working happens to "him", rather than to me - "him" is the object of the verb work. In English, the subject of both sentences remains the same - "I" is in the nominative case, and is felt to be unaffected by the direction of the action of the verb. English is a "nominative-accusative" type language, and so the subject of the sentence can have only one form.
But in a language that displays ergativity, the word "I" would have two different forms in these sentences: with the intransitive verb, "I" would be in a form called the "absolutive" case, but with the transitive verb, "I" would be in the "ergative" case. English is not an ergative language, so we use the same form "I" in both examples, but in many of the languages of the third world, these two forms of "I" would be quite different from each other.
Languages that exhibit "absolutive-ergative" grammar are considered to be a different type from languages that exhibit "nominative-accusative" grammar. There are also a number of languages that are "split", in that they exhibit both kinds of grammar in differing but specific circumstances. If you are a native speaker of a nominative-accusative language like English or Chinese, these are some of the most fascinating and challenging topics for the study of unfamiliar grammatical thought-processes.
Ergativity is rare in African and European languages, although it seems to have existed in the extinct Anatolian branch of Indo-European, possibly because of language contact. To my knowledge, the three modern ergative languages that are the most standardized and have the largest populations are Tibetan, Basque, and Georgian. The study of ergativity, because it so often involves languages like Dyirbal, Yidin , Yukagir, Jê, and so on, spoken by small populations in pre-industrial societies, is challenging to say the very least. This study depends on good linguistic field-work and is an obvious reason to support the Endangered Languages movement (see also the write-up at Encroachment of major languages endangers biodiversity). The British linguist R. M. W. Dixon, who has made his career in Australia (a veritable hot-bed of ergative aboriginal languages), is a recognized authority, and has written a useful book on the subject published by the Cambridge University Press.
Our verbally fecund colleague Gritchka has posted a far more technical write-up than this one, detailing some of the interesting complexities of ergativity in conjoined relative clauses. See syntactic ergativity.