There are at least a dozen common constructions which use that.

Indicative that: "Which apple do you want?" "That apple." "This apple?" "No, that apple."

Causative that: "Why did you go to school?" "So that I could study Japanese."

Antecedent that, where that takes the place of some concept which has come before: "Did you study Japanese?" "That is a given."

"Never going to see it outside of literary constructions" that: "That which cannot be seen is more real than that which cannot be felt."

Adjectival-clause-introducing that: "We are looking for two students that are studying language pedagogy."

etc, etc

English grammar, fun stuff.

While looking at the Wikipedia entry for definite article, I found that (http://en.wikipedia.org/wiki/Article_%28grammar%29#Definite_...) that was derived from þæt, the neuter singular form of the in Middle English.

Fascinating stuff!

pardon this non-native, but shouldn't "We are looking for two students that are studying language pedagogy." be "We are looking for two students _who_ are studying language pedagogy." or possibly _which_ ? Or are they equivalent?

In this case, 'who' and 'that' are mostly equivalent. If you're talking about a specific person, sometimes 'who' is the only choice, but here it probably doesn't matter.

'Which' wouldn't be used here because 'are studying language pedagogy' is restricting/filtering 'students', and so it's essential to the meaning of the sentence. 'Which' is used for clauses that add non-essential information.

    My shirt that is red is upstairs.
    My shirt, which is red, is upstairs.

The first sentence is contrasting the subject with other shirts (..., but my blue shirt is not). In the second sentence, there's only one shirt being discussed - it happens to be red, but even if that information was left out, it wouldn't change what shirt you were discussing.