Easily my favorite part of Montreal.
The old stone buildings and narrow cobblestone streets give you that very quaint feel of walking through an old European village.
The cafes provide plenty of opportunity for people watching while sipping on coffee or chomping down on your croques-monsieurs. If you manage to get a patio seat at one the restaurants on Place Jacques-Cartier you can even take in a show presented to you by street performers.
Art lovers can give their wallets a good workout at any of the many galleries that line up along Rue Notre-Dame.
The Basilica Notre-Dame is a must see for only $5 and for another $5 you can catch the light show which I hear is very beautiful.
If you tire of walking there are horse carriages waiting to whisk you away on a tour of the old town, the car drivers stuck behind it won't enjoy it so much however. :)
There is so much more to do, including the old port, but the above is a good start.