As GregLocock pointed out, it's all about economics. Aircraft are made primarily of aluminum for the same reason cars are still made from steel. It gives you the most bang for the buck.
However, "conventional" aircraft are not just made entirely of aluminum alloys. There are also components made of steel alloys like bearings, gears, linkages, landing gear and fasteners(4340, 9310, MP35, AerMet 100, etc), turbine blades and fasteners made from nickel alloys (Inconel 718, 15-5PH, A286, etc), structures, compressor blades and hi-loks made from titanium alloys (6Al-4V), wiring and bushings made from copper and copper alloys (BeCu 172). Each material is carefully selected for a given application, but in the end it is always a compromise between various conflicting requirements, such as weight, strength, raw material cost, fabrication cost, material availability, etc.
The weight-to-cost implementation trade-off for a production commercial aircraft like a 737, used to probably be somewhere around $300/pound. With current high fuel costs, that number is probably much higher. And that's why the new generation of commercial aircraft like the 787 are using lots of (very expensive)carbon composites and titanium. Even older aluminum structured aircraft, like the 737, used titanium for all of its permanent fasteners (ie. hi-lok's), because even at $40 or $50 per pound, titanium was still attractive for its weight savings.
A hard lesson that Boeing has just learned with their 787 is that even though you can engineer and prototype an aircraft in rare and exotic materials like carbon composite and titanium, you must also ensure that you will have a reliable, consistent and stably-priced supply of those materials for the duration of production. If you doubt me, just check with your local supplier for delivery lead-times of titanium fasteners or carbon pre-preg.