Shakespeare's Romeo and Juliet is a much loved classic. Most people know that the tragedy that results in the deaths of Romeo and Juliet and that is anticipated, even in the Prologue to the first act, stems from the "continuance of their parents' rage" (Prologue.10). The prologue even suggests that the feud between the families is so entrenched that only something as tragic and absolute as the death of these "star-crossed lovers" (6) could end this "ancient grudge" (3).
Both the Montague and the Capulet families are well-respected in Verona but the deep-seated hatred between them even extends to the servants. Samson, a servant of the Capulet household, claims in Act I, scene i that even though he may not be inclined to fight, when it comes to "a dog of the house of Montague" (8), he will "show myself a tyrant" (21). Tybalt, Juliet's cousin, expresses a hatred so unshakable that he may not even be aware of the actual cause of the rift and would appear not to care. He is hot-headed and passionate and compares his hatred of the Montagues to hell itself. Tybalt's death later and at Romeo's hand will end any hope that Romeo and Juliet may have had for future happiness.
The Capulets are having a feast and a servant says to Romeo that he is welcome at the feast as long as he is not from the house of Montague. It is in Romeo's best interests to go because Benvolio wants Romeo to get over his infatuation with Rosaline and see for himself that there are many other beautiful ladies who deserve his attention. As Mercutio, Romeo's friend, has been invited, Romeo will accompany him and, upon seeing Juliet is prepared to renounce the name that is "hateful to myself" (II.ii.55). Juliet is so in love with Romeo that she is also prepared to give up her name and says, "I'll no longer be a Capulet" (36). Juliet speaks the often quoted line, "What's in a name?" (43).
The two "houses", as Shakespeare calls them, are the Capulet and Montague households. Romeo is the son of Lord Montague, and Juliet the daughter of Lord Capulet. Tybalt is a cousin to Juliet - Lady Capulet's brother's child. Both families lose their heirs at the end of the play - the younger generation is wiped out.
Just to clarify, Paris and Mercutio are kinsmen of the royal family - relations of the prince, and do not belong to either family; though, by the end of the play, the two do end up dead.
Romeo comes from the Montague family and Juliet is a Capulet. The reason for the long-standing feud between the two families is not given, but it also causes the deaths of two other family members, Mercutio and Tybalt. The two families are reconciled in the end.