How is the feud between the Montagues and Capulets depicted in Act I, scene i of Romeo and Juliet?

The Montagues and Capulets are two of the most distinguished families in the city of Verona, the geographical setting of the play. The two families are equal in power and authority. Since these two households hold such prominent positions, their actions influenced the general day-to-day affairs of the city. Their behaviour obviously also piqued the interest of both the general populace and people occupying rank. Furthermore, their influence would extend well beyond their immediate families because they would also have loyal followers and subjects. As such, they would probably be the subject of much gossip. 

This is, essentially, the situation we are confronted with in the opening scene of the play. We are informed in the prologue that these two esteemed families have been caught up in an "ancient grudge." They have become lifelong enemies. The reason for this animosity is never explained, but it seems to have existed for generations. At the time of the action in the play, the Montagues' and Capulets' animosity has flared up and created all sorts of disruptions in the city. Their fight has spilled out into the streets, disturbing the peace.

Two households, both alike in dignity,
In fair Verona, where we lay our scene,
From ancient grudge break to new mutiny,
Where civil blood makes civil hands unclean...

The battle between the two families has driven ordinary law-abiding citizens into committing crimes, either because of their loyalty to one or the other family or in self-defense, since, as spectators, they could be affected by the actions of the brawling combatants. The blood of innocent bystanders has clearly been spilled during these altercations and, therefore, the Montagues and Capulets would be criminally liable -- their hands have been tainted with blood. 

The prologue clearly indicates the attacks were vicious and bloody, not mere verbal arguments. The depth of the two families' hatred for each other is distinctly conveyed.

It is this deep loathing for one another that forms the backdrop to our story, for the two protagonists, Romeo and Juliet, are each children from an opposing family. Fate has played a cruel trick on the unfortunate two, for they fall desperately in love even though their union is doomed to fail. The prologue states:  

...From forth the fatal loins of these two foes
A pair of star-cross'd lovers take their life;
Whose misadventured piteous overthrows
Do with their death bury their parents' strife.
The fearful passage of their death-mark'd love,
And the continuance of their parents' rage,
Which, but their children's end, nought could remove,
Is now the two hours' traffic of our stage;...

Act I, scene i introduces us to the conflict between the families in the form of a conversation between Sampson and Gregory, two men loyal to the Capulets. They are discussing the feud and their involvement therein. Both men are clearly prepared to risk their lives for the family they serve. The two are soon confronted by Abraham and Balthasar, men from the opposing Montagues. A verbal altercation ensues in which the men hurl insults at each other. This develops into a fight between Sampson and Abraham.

Benvolio, from the house of Montague, turns up and tells the two combatants to lay down their weapons because they do not know what they are doing. He beats down their swords, stopping the fight. Tybalt, from the Capulet house, enters and challenges Benvolio. The two men start fighting and soon other members of the opposing houses get involved. Citizens take up clubs to beat up those from each house and soon there is a huge brawl in the streets.

Soon the leading members of both families arrive and intervene. The prince of Verona also makes his appearance, and his stern admonition brings the fight to a close:

...On pain of torture, from those bloody hands
Throw your mistemper'd weapons to the ground,...

The prince proceeds to address the heads of both households:

Three civil brawls, bred of an airy word,
By thee, old Capulet, and Montague,
Have thrice disturb'd the quiet of our streets,
And made Verona's ancient citizens
Cast by their grave beseeming ornaments,
To wield old partisans, in hands as old,
Canker'd with peace, to part your canker'd hate:
If ever you disturb our streets again,
Your lives shall pay the forfeit of the peace.
For this time, all the rest depart away

The prince is extremely upset at the regularity of these public battles and warns the Montagues and Capulets will face execution if they disturb the peace again. The prince then asks everyone to leave and requests that Lord Capulet accompany him and that Lord Montague visit the prince later. The prince obviously wants to speak about the incidents and ask the families to resolve their differences and bring peace to Verona.

It is, therefore, tragically unfortunate that the prince's intervention does not have the desired result and many become tragic victims of this unnecessary feud later in the play. Those victims' deaths, though, finally bring the conflict to an end.