Why did Hermia and Helena quarrel in A Midsummer Night's Dream?

The answer
to this question depends on which part of the play you are referencing.

At
the beginning of the text, Helena reveals that she is in love with a man named Demetrius who is
romantically interested in Helenas friend Hermia.

Hermia, however, is in love
with another man named Lysander, although her father has arranged her marriage to Demetrius.
While this does not necessarily constitute a quarrel, there is an underlying conflict between
the two women, since one of them is left alone without a suitor.

Later on in
the play, Hermia and Helena do engage in a verbal spat. This occurs after Puck mistakenly causes
Lysander to become infatuated with Helena. Under this spell, Lysander denies that he ever loved
Hermia. This causes Hermia to become angry with Helena, whom Hermia accuses of bewitching
Lysander.

Unable to believe that anyone could be in love with her, Helena is
perplexed by both Lysanders and Demetriuss professions of loveboth of which were given under the
influence of artificial love spells cast upon the men by the forest fairies. As a result, Helena
thinks that both men are making fun of her, and she accuses Hermia of joining in their
mockery.

Although this quarrel is resolved, the two women experience a test
of their friendship in the forest outside Athens. Each of them insults or mistrusts the other
because she lacks the full explanation of what occurs with the two male suitors. Ultimately,
their friendship is restored once the truth is revealed and the marriages secured at the end of
the play.