Read the full text of Henry VI Part 1 Act 1 Scene 3 with a side-by-side translation HERE.
The Duke of Gloucester, Lord Protector of the realm (more on that here) tries to enter the Tower, afraid that things may have gone wrong there since Henry V's death. The warders refuse to let him in, saying they have orders not to.
Gloucester is understandably annoyed by not being let in when he's basically (and legitimately) running England, but when the guy in charge comes (he's named Woodville), he says that he has clear orders not to let Gloucester in. This is sort of like locking Dumbledore out of Hogwarts. Gloucester isn't quite as full of professor-ly charm as Dumbledore, but still.
Worse yet, Woodville says it's Winchester who's ordered them to keep Gloucester out. As we know from the very first scene of the play, Gloucester and Winchester don't get along.
Gloucester disses Winchester badly—he says Henry V never liked Winchester anyway, and that Woodville had better open the gates.
Winchester turns up about now and starts things out by insulting Gloucester. How so? He calls him "ambitious." Now that sounds nice—ambition means you accomplish great things and get your homework done early, right? Well, not in the Renaissance, especially if you're a nobleman but not a king. People suspected ambition and thought an ambitious character might want to take over.
Gloucester disses Winchester right back, and the insults get thick and fast. Winchester accuses Gloucester of usurping the King's power and betraying the kingdom, and Gloucester accuses Winchester of trying to murder Henry V way back when, among other things.
The insults keep going until a fight breaks out, and Gloucester's men chase Winchester's men off.
The Mayor of London comes and asks why they can't all just get along. Gloucester and Winchester tell him, at length. Gloucester complains that Winchester has no regard for God or King (pretty bad insults for a member of the nobility and a clergyman in this time), and that he's taken over the Tower. Winchester rattles off a list of accusations against Gloucester: He keeps pushing for war, he asks for too much money, he wants to overthrow religion completely, and he wants to betray Henry VI and be king himself.
Gloucester is ticked off by this list, unsurprisingly, especially since lots of it seems to be false. He says he'll answer with blows, and the fighting starts again.
The Mayor gets one of his officers to make a loud proclamation, which basically says, "Go home, and now you can't use weapons in my city." It even threatens the death penalty if they do use weapons here again. Dude's laying down the law, old fashioned sheriff style.
Gloucester and Winchester agree to stop fighting now, but only on the condition that they can keep feuding in other ways.
Gloucester is more moderate: He says he won't break the law, but that their business isn't over.
Winchester is less moderate: He basically threatens to kill Gloucester at some later point.
The Lord Mayor says if they don't stop he'll bring in clubs. He apparently doesn't like Winchester either, since he mutters to the audience, "This cardinal's more haughty than the devil" (1.3.85). Burn.
Gloucester acknowledges the Mayor's authority, and Winchester makes another threat on Gloucester's life. They leave.
The Mayor marvels that the nobles could be so quarrelsome.