It is one of those words like ‘set’. It can be a noun, verb, adjective or adverb. The reason you don’t find it in any dictionary is because the lexicographists (or whatever) are too lazy to list out all its meanings and connotations. The reason it is not used very widely is that most people are not able to comprehend its profundity, and hence are afraid to use it.
The most commonly used form of the word is as a noun, more specifically as a proper noun. It is a quite common first name, especially in North India. Famous people with the name include that loser hero in Sholay who used a coin with heads on both sides so that he could die, thus avoiding marrying a widow! Then, there was this supposed “excellent all-rounder” called Jai P Yadav, unfortunately for whom Souravda got sacked as captain at the wrong time.
Another noun form in which the word is used is in a context such as “jai happens”. I’m not too sure though whether this should be classified as noun or adverb (could someone good at English enlighten me?). “Jai happens” is one of those really unmitigated sentences which manages to capture a lot of meaning in just two short words. This phrase was thus invented so that one could save the trouble of saying a whole paragraph when one could use just two words. It is for this very reason that I can’t explain the meaning of this phrase here. Guess it would suffice if i say that it is a very potent phrase.
Then, there is the commonly used verb form, where it is approximately synonymous with death. Something like “Rajiv Gandhi jai” implies Rajiv Gandhi is no more. Another common usage as a verb is “FTSA Assignment Jai” which means that the FTSA Assignment is so tough that you’re giving up on it. Once again, note that it is not synonymous with death, it is much more insightful than that. The whole reason for the word to exist.
This time, I’m sure that it is an adverb when I use it in the context of “Atticus Finch told jai”. Note that in certain contexts it can be used interchangeably with “Atticus Finch jai”. However, this is a restricted usage, as “Atticus Finch told Jai” has more connotations than just Atticus Finch dying. The most commonly used context where you say that is when Atticus Finch was supposed to meet you, probaby to team up with you in a quiz, or to give you money and suddenly decides not to turn up.
The least used form of this word, however, is as an adjective. It can be used in the context of “this is a Jai assignment”, where jai stands for “extremely tough” or “killing”. However, this is so complex a form that very few people have actually managed to understand the word in this context, thus leading to the low usage.
I have tried my best to explain to you the great insights that this simple three-letter word offers but I must confess that the English language is hardly sufficient to delve into the intricacies of it. I tender my humble apologies for the same. Oh no, I had almost forgotten to tell you one of the most important (and popular) usages of this word.
Bhaarat maata ki jai!
Jai in “jai happens” is a noun.
Not to be an asshole, but the constant invention and ostentatious usage of slang by a lot of yuppie Indians strikes me as a signaling mechanism. Its point seems to be to advertise membership of a select (or at least self-perceived select) group. The corollary is that once there is widespread adoption of a particular slang, people feel the need to invent new slang to differentiate themselves once again from the uncool masses.
Linguistically or communication-wise, there is little point to the slang. I do not claim that it makes the speakers more stupid (I am not that kind of a language maven), but there are simpler, more unambiguous, and probably more elegant ways to communicate than to use such slang. This gives me all the more reason to suspect that slang usage is signaling – a signaling mechanism needs to be costly for it to be believable, otherwise it will be too easy to fake (think peacock’s tail or a deer’s stotting). Slang usage can be made costly by overloading the meaning of the slang, making the usage itself grammatically unconventional, and by making the meanings counterintuitive.
Btw, I am not dissing or criticising slang usage by saying all this. Signaling explains a lot of human behaviour, and this just seems to me to be another instance of it.