This is all off the top of my head but should give you an idea:
Within the city, it is the areas around Hillcrest Park (especially along High, Summit and Winnipeg avenues), the area along Court Street between River and Clavet (especially on the west side of the street, which has impressive views of the lake) in Port Arthur, and the area bounded by Arthur, the CN Mainline, Walsh, and Syndicate in Fort William. These are the "old money" neighbourhoods (though they don't have much old money left), the houses are impressive but many are multi-unit now. There are still a lot of upper middle class families in those areas though, and they are more well off than almost everyone else in the city.
The richest people in the city live in in the subdivisions along Highway 61, south of the Kam river, in estates throughout the rural northwest part of the city (especially along Oliver and Government roads), in the unincorporated area of Lappe/Kaministiquia (aka Gorham and Ware), and along the lakeshore in Shuniah and Neebing.
If you're considering buying, the options are: North and northwest of the city, a middle-of-the-woods pioneer setting with a mix of people and professions. Good hunting/nature watching. Cost: $$/$$$$$ Average Income: $$/$$$$$ Scenery: 2/5 (boring unless you like trees or score a lake lot)
Southwest of the city, a lot of people in the business field (lawers, doctors, prominent businessment), basically the area plays the role the historic old-money neighbourhoods played 100 years ago. Probably good business connections there. Lots of golfing. Cost $$$ Average Income $$$$ Scenery: 5/5 if you get a mountain view. 1/5 if you don't.
Along the lake and in Silver Islet, rustic lake shore retreat, but its getting crowded. Mostly retirees. Cost $$$$ Average Income $$$ Scenery: 6/5 best in central Canada
Fort William's Inner city old money historic mansions area: mix of people and professions, but very attractive neighbourhoods if you like an urban setting. Notable residents include regional politicians, lawyers and doctors, and university students renting multi-unit conversions. Increasing senior population. Nice buildings and tree lined streets, but poor people live near by and there is a high school in the neighbourhood. The areas along Arthur and Syndicate are blighted but improving. Cost $ Average Income $$ Scenery: 2.5/5.
Port Arthur's old money areas: mix of people and professions, but very attractive neighbourhoods if you like an urban setting. Notable residents include regional politicians, lawyers and doctors, and university students renting multi-unit conversions. Increasing senior population. Nice buildings and tree lined streets. Adjacent to less wealthy neighbourhoods. The south end of the Hillcrest Area (near John Street) isn't very attractive but is improving. The area around the intersection of High and Beresford near Hillcrest Park is the most sought after part--houses valued at well over a million but sell for around 600,000. Cost: $$$ Average income: $$$ Scenery: 5/5 if you have a lake view, 2/5 if you don't.
Here is a map to give you an idea of where this stuff is located:
http://img440.imageshack.us/img440/4784/image2993.png