In Act III, Jack finalizes the play with the sentence
I have finally realized for the first time in my life, the vital Importance of being earnest.
By extrapolating the phrase "being earnest" several ironies come to light.
There is not a single character in the play named Ernest. Hence, in saying "earnest" Jack is using the word as an adjective (honest, real) and as a homophone for the fake "Ernest". Since the entire action of the play was fueled by "someone" being called Ernest, and such person is a fake, the entire phrase is ironic; almost laughable.
In that same sentence, when Jack specifies the words
the vital importance of being earnest.
there is another major irony: since nobody is really being earnest in the play, there is no vital importance. The entire Ernest charade only became real by accident. The convoluted story of Jack results in the ironic coincidence that his name, by right, is indeed Ernest and so he had been telling the truth the entire time. The paradoxical turn of events leads Jack to realize that, although there is no earnestness intended, there was earnestness realized after all.
Therefore, what Jack really tries to say with the phrase is that, for the first time, he sees that it may be a good idea to tell the truth from the start, after all; that being honest is a good thing. However, the audience still is left in doubt as to weather Jack meant this or if he actually meant that it is important just to be "an" Ernest for the sake of the many events involving this fake character.