During World War 2 Rome was declared to be an "open city" but I do not know what this means. What rules are there in an open city? Who controls it or is it neutral? I want to know what happens when a city is declared an "open city".
If you google Open City, you'll find the answer
In war, in the event of the imminent capture of a city, the government/military structure of the nation that controls the city will sometimes declare it an open city, thus announcing that they have abandoned all defensive efforts.
The attacking armies of the opposing military will then be expected not to bomb or otherwise attack the city, but simply to march in. The concept aims at protecting the historic landmarks and civilians who dwell in the city from an unnecessary battle.
An open city is not contested; the city government declares that they have no intention of resisting the occupier.
Basically that the city has abandoned defense. It was done because of the history of the city and the desire not to have it bombed.