I want to know, what are the differences between "in the USA" and "in USA"? When to use "the" before a nation's name?
You would say "in the USA" (although it's worth pointing out that you would say "in America," not "in the America").
As chrylis pointed out, the word the is linked to the word States. So, to answer the question, "Where was this car made?" (assuming the car was made in Detroit), one could say any of the following:
It was made in the United States.
It was made in the States. (somewhat informal, but acceptable, esp. conversationally)
It was made in the USA.
It was made in America.
It was made in North America. (less specific; refers to the whole continent, but still a valid answer)
It was made in the Americas. (even less specific, refers to two continents, plus Central America, too)
Getting back to your original question (when to use the when referring to a nation), that happens when one of the key words in the country's name requires an article. This could be because of a word like Kingdom or Republic:
I bought this in the Republic of Korea. (but: I bought this in South Korea.)
I bought this in the Dominican Republic.
I bought this in the UAE.
or because of plurality:
I bought this in the United States.
I bought this in the Philippines.
I bought this in the Bahamas.
Be careful, though! Not all countries ending with "s" are plural:
I bought this in Honduras.
"In the USA" is correct.
We put "the" in front of the names of some countries, though for most we do not. Basically we use "the" in front of names that are "descriptions" rather than simply a name, like "the United Kingdom" and "the United States". We don't use "the" in most cases where it's plain name, like "France" or "Latvia".
Some countries have "long forms" of the name that add descriptions and so call for "the", but also have a short form where we do not use "the". For example, "I visited China", but "I visited the People's Republic of China". Note no "the" in the first case but we do use "the" in the second case.
There are a few countries with names that don't sound like descriptions but that do use "the". Examples I can think of off the top of my head are "the Bahamas" and "the Philippines". We used to say "the Ukraine", but I understand that nation has said that it prefers to simply be called "Ukraine". I've heard both "Gambia" and "the Gambia". You just have to learn these special cases by rote.
"In the USA" is correct. America is composed of states and 'states' is a noun. You can address it as America. For example, you can say that "ABC lives in America". But to be more specific,
ABC lives in the United States of America
looks more complete.
Although other responses are correct, if we want to decompose the word USA into its meaning, I would say that "USA" is a designation of "The United States of America" like "BMW" is a designation of a unique name.
So you can say "Made in USA".
But it's just my opinion.
USA is an abbreviation for United States of America, and the is used because the proper noun is a noun phrase whose main noun is the ordinary noun States. The definite article is also used in other country names (and their abbreviations) that are phrases made from ordinary nouns, such as the United Kingdom or the People's Republic of China.
Other languages have similar usage; the official name of Argentina in Spanish is la República Argentina (the Argentine Republic), and the country is referred to in Spanish as la Argentina.