Go ask his name.—If he be married,
My grave is like to be my wedding-bed.
His name is Romeo, and a Montague,
The only son of your great enemy.
My only love sprung from my only hate!
Too early seen unknown, and known too late!
Prodigious birth of love it is to me
That I must love a loathèd enemy.
At her father's masked ball, Juliet falls in a big way for the disguised Romeo, a Montague and thus an enemy of her family [see A PAIR OF STAR-CROSSED LOVERS and DANCING DAYS]. Even though she has nothing personal against the Montagues, Romeo in particular, she can't escape being a Capulet, or escape her family's "hate."
Today "prodigy" usually refers to a precocious youngster. But the word had much different connotations in Shakespeare's time: a "prodigy" was someone or something abnormal, a monstrosity. Prodigies were taken to be omens of a family's bad fortune; this idea was thus naturally linked to the idea of birth, an event surrounded with a vast structure of superstition. While the word "prodigious"—of sixteenth century origin—was often applied to newborns or children, "prodigious birth" seems to be Shakespeare's coinage. When Juliet refers to the "prodigious birth" of her love, her imagination runs through the horrors such an omen portends; with deadly dramatic irony, she foresees that her grave will in fact prove her marriage bed.
In a play contemporaneous with Romeo and Juliet, Shakespeare has a fairy-king bless the marriage beds of three newly-wed couples. He commands that "the blots of Nature's hand/ Shall not in their issue stand;/ Never mole, hair-lip, nor scar,/ Nor mark prodigious, such as are/ Despisèd in nativity,/ Shall upon their children be" (A Midsummer Night's Dream, Act 5, scene 1, 409–414). This passage is an excellent rendering of Elizabethans' dread of prodigious births.