Most languages have either no articles, or one or both of the definite (akin to English "the") and indefinite (akin to English "a" / "an").
But are there other kinds of articles, and which languages have them?
Yes there are!
Another common one is the partitive. If you've ever learned a language like French then it's an article you use with mass nouns like "water" or "cake". In English the closest would be something like "some" as in "Would you like some cake?". Occasionally you'll see it written by linguists as "sm", as it is phonologically reduced in speech.
Here are some examples from French:
- Je ne bois pas de café.
- Je n'ai pas de temps à perdre.
- Donne-moi de la glace.
I'm sure there are probably others!
It is worth noting, I think, that "article" is not a theoretical primitive in (most if not all) contemporary generative theories of syntax. A generative syntactician would say that languages have larger or smaller inventories of determiners (e.g. this, that; this is the category from which the developed) and quantifiers (e.g. one, many, no; this is the category from which a(n) developed). However, articles are a notion of classical grammar and are definitionally restricted to the definite and indefinite.
(Some grammarians might include more kinds of articles, as the Wikipedia article on articles does. But this reflects more a taxonomic whim of the analyst than a fact about language.)
By some accounts, "no" is considered an article, distinct from definite and indefinite. To me it affects the reference in a very different way than the other two ("a couch," "the couch," "no couch"), but I know none of the proper linguistic theory on the matter.
Don't forget the demonstratives. Old Church Slavic had a definite article (i), a proximal or 'this here' demonstrative (sej), a distal or 'that over there' demonstrative (on), and a simple 'this/that' demonstrative (tъ).
Some languages also have a zero article (i.e., they have overt articles, but can omit them). In English, for example, you can say "I saw turtles on the beach", whereas in French you would have to add the indefinite plural article: "J'ai vu des tortues sur la plage".
As Bill Sullivan noted - The demonstratives in Slavic languages can evolve to articles. One example is Macedonian (and perhaps western Bulgarian dialects):
One feature that has no parallel in any other standard Balkan language is the existence of three definite articles pertaining to position of the object: unspecified, proximate (or close) and distal (or distant).
Romanian has something called Genitival article (which is nothing extraordinary - just an article in genitive)
The wikipedia article about "article" lists: definite, indefinite, partitive, negative.
Catalan has personal articles used with person names, e.g., en Pere "Peter" as opposed to el/un noi "the/a boy".
I just came across some info trying to learn a bit about Albanian in Albania.
Albanian has a kind of article called an "adjectival article".
I haven't been able to find out much about them though so I've asked a question about them.
Creoles tends to have three ways of marking noun phrases for definiteness:
definite, for already known, specific information ({the} in English)
indefinite, for new, unknown but still specific information ({a/an} in English)
not specific, usually unmarked (also {a/an} in English, or unmarked)
The first time you mention something you mark it indefinite, later you use the definite marker and if it is not specific or important you use the third option. Naturally I can't remember the reference for it (book used in grammaticalization-class I think) but apparently there can be two types of nonspecificness too, for new and old nonspecific information: In such a system "dogs bark but foxes bark too" the nouns would be marked differently. IIRC no more than three of the four possibilities are marked in any one language though.
Then there are the languages with only one article of course... or none, but opinions differ on whether all languages always mark definiteness (needn't be with an article!) or not.
For those interested in definiteness there's the book "Definiteness" by Christopher Lyons. I haven't read it myself as I'm not very interested in the subject. The other books I have read in the series it is part of though, Cambridge Textbooks in Linguistics, have never disappointed even though of course some linguists just can't write and needs to be read in small doses lest your eyeballs bleed ;)
Basque has a definite article that could be taken as special in some ways:
-It is an affix -There are no nude Basque NPs, and the affix definte article is added to the NP is no other determiner or quantifier appears -The so-called definite article also appears in predicates, i.e. in phrases with no referential value.