An allegory is a work of literature with a hidden meaning beneath the surface of its narrative. On the surface, Arnold's poem is about a merman, a mythical, pagan figure, who marries a human named Margaret. They have children together and live a lovely, seemingly magical life in the caverns by the sea:
Sand-strewn caverns, cool and deep,
Where the winds are all asleep;
Where the spent lights quiver and gleam ...
But this life is interrupted when, at Easter, Margaret's human side is pulled back to the church and Christianity so that she abandons the merman and his children.
This story is an allegory for what Arnold felt was the coldness and loss of faith in the Christian church at that time, as its traditional ideas were being shaken and tested by advances in science. By going to what is three times called the "little grey" church, Margaret abandons what is vital in life to pursue a small, worn-out ("grey") faith. Even when her husband comes to her and pleads with her to come back to her children, who "moan," Margaret stays stuck, her eyes "seal'd to the holy book."
Margaret symbolizes, or becomes an allegory for, the people who miss life by sticking with a weak, fading religion with little left to offer. The children sing of the loss of their mother, calling her "cruel" for abandoning them.
As he expresses in other poems, Arnold would have liked the Victorian age to have the robust, communal faith he imagined in the Middle Ages, but he saw contemporary Christianity as leaving people lonely and cut off from each other, just as Margaret is cut off from her former life.