O mother-heart! when fast the arrows flew,
   Like blinding lightning, smiting as they fell,
   One after one, one after one, what knell
Could fitly voice thy anguish! Sorrow grew 
To throes intensest, when thy sad soul knew
   Thy youngest, too, must go. Was it not well,
   Avengers wroth, just one to spare? Ay, tell
The ages of soul-struggle sterner? Through
The flinty stone, O image of despair,
   Sad Niobe, thy maddened grief did flow
In bitt’rest tears, when all thy wailing prayer
   Was so denied. Alas! what weight of woe
Is prisoned in thy melancholy eyes!
What mother-love beneath the Stoic lies!

So, I know the original story of Niobe. She was a noblewoman with 14 children and challenged the Goddess Leto by making fun of the fact that Leto only had the twins, Apollo and Artemis. Leto is thus offended and has Apollo and Artemis take their bows and kill all of Niobe’s children. Niobe’s father then jumps off of a cliff and Niobe is turned into some sort of statue or plant (please let me know if I got any of the story wrong).

The part at the beginning, “O mother-heart!” to “what knell” is when Niobe’s children are murdered, arrows through the hearts. All of them dying. The “Could fitly voice” to “O image of despair” describe Niobe’s anguish and her begging that at least her youngest should live, that they don’t deserve to die, but they still do. The “Sad Niobe” to “weight of woe” is obviously a description of Niobe’s grief of having all of her children die.  Then the last to lines are most certainly Niobe being transformed into some form of statue with the whole “is prisoned” line.