The neighborhood closely corresponds to the original boundaries of the city of Montréal (founded as Ville Marie) as it existed throughout the 17th and 18th centuries. There is even a specific place in the neighborhood (the Pointe-à-Callière, today a history museum) which is believed to be the city's birthplace.
The easiest way to think of Old Montreal (Vieux Montréal) nowadays is as of a tourist ghetto. This is probably too harsh of a description, especially considering the almost two decades' worth of efforts to turn the neighborhood into an authentic place of urban living, but nevertheless it works as a rough mental shortcut. Despite the aforementioned efforts, the neighborhood empties out in the evening (except in the summer) and lacks real vitality; most of its commerce is geared toward tourists. Old Montreal lacks supermarkets (though it has convenience stores), laundries, gyms, and similar amenities. Hardly any restaurant here can be called a "neighborhood joint." On the other hand, souvenir shops (Rue Notre Dame) and art galleries (Rue St-Paul) are plentiful, and some of them even stay open late – in case you ever find yourself in urgent need of something to decorate your wall with.
Sarcasm aside, it's not all bleak. For starters, the historic center has many beautiful buildings, many of which, after years of neglect, have been renovated and/or converted into residential use as lofts (a handful are available on AirBnb and similar platforms). Secondly, there has been much development along the outer edges of the neighborhood – not great development, mind you, just decent, but which has nevertheless vastly improved the neighborhood's local/tourist ratio, by osmosis. In particular, Rue McGill, on the western side of Old Montreal, has seen plenty of activity and as a result, now has several restaurants targeting locals. Thirdly, most of those urban conveniences that Old Montreal still lacks can be found within walking distance: for example, the IGA supermarket on the Boulevard René-Lévesque can be reached on foot in 15-20 minutes from almost anywhere in Old Montreal.