The word Jain came from “Jin”. We are the followers of “Jin” so we are called Jain.
“Jin” means one who have achieved Jeet (Victory) over internal and eternal enemies including Karmas. They are the Enlightened Souls who have achieved Omniscience (Pure and complete knowledge – all knowing, all seeing, can observe past present and future of the entire universe; Keval Gyan).
They have showed us the true path to achieve salvation.
By saying Jai we are praising the glory of those great Souls (Tirthankaras) and by doing this indirectly we are trying to internalise and imbibe their greatness in ourselves. By Jinendra we are saluting and appreciating the virtues of those Jins who also have achieved full control over their Indriyas (Senses).
Every Jain or on fact everyone should always start the communication (both written and oral) with “Jai Jinendra”. By admiring the greatness of these Emancipated Souls we get benefitted and feel more purified at the same time achieving further victories over our Karmas.
Jai-Jinendra” or Jai (Jaya) Jinendra:
It has two parts Jaya (most of us writes Jai) and Jinendra.
Let us look at Jinendra word first. Jinendra word is made of two words: 1)Jina + 2) Indra.
As we all know Jina means the one who has completely conquered the passions like attachment and hatred.
As per Sanskrit Dictionary by Vaman Shivram Apte, Indra means the lord of gods.
Therefore, Jinendra would mean Lord of Jinas (for example: Narendra means lord of men). You might know that there are two kinds of Jinas: 1) Ordinary Jina and 2) Tirthankar Jina.
There is only one Tirthankar at a time while there can be numerous Ordinary Jinas at the same time. Tirthankarhood is acquired only by performing special kind of austerities in a particular fashion. Therefore, Jinendra would mean Lord of Jinas, The Tirthankaras.
As per Sanskrit Dictionary by Vaman Shivram Apte, Jaya means Conquest, triumph, victory. When you explore more on triumph, you find theses words: commemorate, pay homage to, admire, honor, venerate, etc.
So if we put all of theses words together, the meaning behind saying “Jai-Jinendra” means Honor to Tirthankaras, paying homage to Tirthankaras, or admire Tirthankaras, etc.