It's been discussed a few times, on this forum.
I'll give it from a Christian perspective, though there are many other ways that this could be expressed.
In my own case it was a "special experience", but that's not always the case.
It's basically what Christian groups call being "born again" (which is an event, or point in time, not a denomination). In fact it's not limited to denomination, or even religion at all. Hence (IMHO) the dogmatic will often stifle their own growth.
Some just come to a dawning realisation that Christ is special to them personally, and that they have have a relationship with him. They often can't put a day or hour on this, but "just know".
Others, like myself, have a "Damascus Road" experience, where we can pinpoint the minute.
The bottom line, from a Christian perspective, is that it's personal and individual. No two experiences are the same, which makes it even more difficult for canned-spirituality JW's to accept. The common denominator is Christ.
It involves something tangible in it's own right, called "faith", which is not subject to the many definitions that have often been bandied around. It's not blind, belief, trust, hope, or many other things that already have a definition in their own right.
Portions of the bible (and in some cases, other works) begin to read like a personal love letter, with meaning springing out.
But even yet, each individual is on a road that is tailored for their own understanding.
I would simply put it another way - it is the point at which the individual becomes consciously connected with the Divine.
At this point life is never the same again. It is incredibly humbling, to touch the infinite.
It is something that was always there, accessible to all, yet elusive. However, once contacted seems so simple as to bring wonder that it was never before seen.