I’m interested in contributing to the canonical post. As a non-native speaker, the English articles have always been a trouble and I can understand why non-native speakers cannot wrap their heads around them.
If we are really going to write a canonical post, I would like to suggest a few topics that might be worth talking into account.
When talking about articles, we'll inevitably be talking about noun and noun phrases (and by extension countability, definiteness, identifiability etc.). Therefore, I believe the best place to start is the basic structure and the semantics of the noun phrase (NP).
BASIC STRUCTURE OF NP
An NP in general comprises a head noun and followed or preceded by modifiers. Usually there's only one head, which is virtually always a noun. The modifiers can be adjectives, relative clauses, or prepositional phrases. An NP usually has a determiner, which always precedes rather than follows the head noun. The tree diagram below shows the basic structure of an NP. This is from Huddeston & Pullum (2005), which has its own way of labeling each node. Every node is labelled with both function and category:
An NP has a special slot for determiners, which have their own structure and ordering. In some grammars (eg. that of Quirk et al., 1985), determiners may be divided into three categories, namely, pre-determiners, central determiners, and post-determiners. The figure bellow shows the basic structure of the determiners:
Articles fall within the central category. Central determiners are mutually exclusive; therefore, *the an apple is ungrammatical.
THE SEMANTICS OF NP
According to Dixon (2005: 82), there are at least three types of meaning associated the grammatical class of noun: viz. CONCRETE – girl, horse, wrist, ABSTRACT – time, future, yesterday, STATES – pleasure, joy, strength, ACTIVITIES – decision, war, game, and SPEECH ACTS – question, order, report.
Semantic classification of noun is important because it can determine the countability of the noun. For example, STATE nouns seldom have a plural form and therefore do not normally occur with the articles ‘a’ or ‘an’, eg. *a jealousy.
The definite article The basically pre-empts the ‘which’ question. It indicates that the head of the NP is considered sufficient in the context to identify the referent. Look at these sentences from Huddleston & Pullum (2005),
i [The President of France] has appointed a new prime minister.
ii Where did you put [the key]?
In [i], there can only be one person to become President of France at a given time. Therefore, it uniquely identifies that person. In [ii], there can be any number of keys that might exist, but the article The is used because it is clear which key is referred to (eg. The car to the key you just told me to unload).