The meaning of the poem is that the things that are most beautiful in life are also the things that have the shortest lives.  He says the first leaves of nature are their most beautiful and that Eden was the most beautiful place to live.  Neither of these beautiful things survived.

I think the clearest illustration of this in the novel is Johnny's death.  He was gold for a brief while, but then died.  On a more hopeful note, it seems that maybe Ponyboy will be different.  Maybe he will actually stay gold.  We get this hope at the end of the book because (instead of going and fighting or anything like that) he turns to writing as an outlet for his emotions.

So Johnny was not able to stay gold, but maybe Ponyboy can.

Frost's poem celebrates the purity of life at its beginning ("nature's first green is gold"), and laments its impermanence ("nothing gold can stay").  When Johnny tells Ponyboy to "stay gold", he is referring to the innocence the greasers once had in common as children.  As they grow into adulthood, they lose that guilelessness and become hardened and jaded under the relentless pressures of poverty, social pressure, family instability, and violence.  Darry, whose dreams of attending college on a scholarship are dashed by the responsibility of caring for his brothers, and Dally, who has given up hope for a better life and descended into a cycle of violence and crime, are bitter examples of this, and the other greasers are not far behind.  Ponyboy is different.  He is sensitive and perceptive, and can still recognize and appreciate the beauty in a good book or a sunrise.  Johnny hopes Ponyboy can keep this ability - that he can "stay gold".

Frost’s short poem “Nothing Gold Can Stay” addresses the fragility of nature, that its cycle is such that green leaves do not last forever: “leaf subsides to leaf” (he puns on the second “leaf,” so that it can be read as “leafs leave”) just as “Eden sank to grief,” meaning just as the pleasures of Eden eventually ended and humanity was thrust into the world where suffering exists. This is the fact of life that Pony learns: good things in life don’t last. In its discussion of the novel, Enotes points out that the poem speaks back to the images of sunsets in the story: “Sunsets are short….But it is possible, Pony proves, to remain true to one's self and thereby ‘stay gold.’”

It refers to Robert Frost's poem "Nothing Gold Can Stay" and has a profound impact on Johnny, who relates it to his own imperiled youth. Later, as he is on his deathbed, Johnny's last words are, "Stay gold, Ponyboy. Stay gold...."

Here is the text of the poem:

Nature's first green is gold, Her hardest hue to hold. Her early leaf's a flower; But only so an hour. Then leaf subsides to leaf. So Eden sank to grief, So dawn goes down to day. Nothing gold can stay.

Basically the meaning of the poem is that innocence is something very hard to hold on to.