When the speaker of the prologue says, "From ancient grudge break to new mutiny," he is describing the relationship between the two families: the Capulets and the Montagues.  These are both old families, long-established in Verona, and the feud between them has existed for a very long time (perhaps for so long that no one really remembers why it even began -- at least, the origin is never explained in the play -- making the violence and deaths of the characters seem all the more futile and fruitless).  Further, the speaker claims that this ancient grudge has apparently led to more recent violent altercations of late.  We can understand, then, that the "grudge" seemed to go away for a while, or, at least, it didn't affect the peace of the community.  It doesn't take long for the audience to identify one of the major perpetrators of this violence when we witness the interaction between Tybalt and Benvolio shortly thereafter.  Therefore, the line roughly translates, in modern-day English, to "From an old feud leading to recent violent fights." 

The phrase you asked about is part of a larger sentence that begins the prologue of Romeo and Juliet.  All told, the beginning of the sentence reads, "Two households, both alike in dignity, in fair Verona where we lay our scene, from ancient grudge break to new mutiny..."  In modern English, this would roughly translate as, "In Verona, two well-to-do families who have a long-standing feud have recently experienced new bouts of violence against each other."  This explains to the audience that the feud is long-standing, and will help to provide some context for the fight between the Montagues and the Capulets (and their servants) that will break out shortly thereafter in Act 1, Scene 1.  This will also help the audience to understand why a relationship between a Montague and a Capulet would prove so problematic.