I have heard several countries expressed in Spanish with a definite article before the country name (e.g. los Estados Unidos, la Argentina, la India). Is there a rule for when this occurs and when it doesn't? Is the definite article optional, or are there certain countries for which it is required? If both forms are used, which is more common?
I am from Argentina and I never say or hear "el Chile", but "la Argentina" is quite accepted and used. The standard explanation of this apparent anomality is that "Argentina" is originally an adjective, tied to the (often tacit) substantive "República", so that the full expression would be "la República Argentina" (analogously to "los Estados Unidos"). Hence, the right thing is to use the article "la", also when the substantive "república" is omitted. (Reference: elcastellano.org). I must admit that this explanation sounds a little artificial to me.
The names of some places, such as "El Salvador", "El Cairo" or "La Haya" always include the article (just as "The Hague" does in English). For some others you can optionally add the article, the RAE lists several examples in the page about the definite article "el".
(el) Afganistán, (el) África, (la) Argentina, (el) Asia, (el) Brasil, (el) Camerún, (el) Canadá, (el) Chad, (la) China, (el) Congo, (el) Ecuador, (los) Estados Unidos, (la) India, (el) Líbano, (el) Pakistán, (el) Paraguay, (el) Perú, (el) Senegal, (el) Uruguay, (el) Yemen, etc
In any case, whenever the article is optional the most common form omits it (and in fact most of these examples sound terrible to my ears, even though they are occasionally heard).