Ergativity

For ergativity, we need to introduce a few other concepts.

Transitivity

A transitive verb is a verb that has both a subject and an object. Examples:

I see her.

Subject in italics, object in bold. The notion of transitivity is not entirely uniform - there's syntactic transitivity, which is pretty much "is there a direct object present (or at least marked for or strongly implied)", and semantic transitivity, which seems way more fuzzy around the corners.

It might seem weird to consider 'I shot it' to be transitive, but 'I shot at it' not to be transitive. Languages may differ in how they understand such constructions - verbs may have adpositions as 'integral' parts, and then 'shot at it' would be transitive; English seems to have some phrasal verbs to be integral, some not. Who knows, really.

Anyways, an intransitive verb is a verb that does not have a direct object.

We can now go on and consider subjects.

Ergativity

Ok, so we have a transitive verb:

I see her.

Let us call these A and P, agent and patient. And we consider what to do when we have a verb that only has one argument - we call it the undergoer, simply to provide a distinction. In English, undergoer is marked the same as agent:

I sleep.

However, in ergative languages, the undergoer gets the same marking as the patient:

I see her. Her sleeps 

Of course, this needn't only be present in the marking on nouns and pronouns, it can also be present in verb congruence and the like. I.e. we may have congruence only for the absolutive (the object or intransitive subject) argument, we may have congruence only for the ergative argument (the subject when there's an object present), thus giving patterns like:

I sees her.
her sleeps.
She am seeing me.
Me am sleeping
But
her is sleeping.
I is seeing her

However, congruence in some languages follow a nominative pattern despite having ergative case:

I see her
her sleeps
me am sleeping
her is sleeping
I am seeing her