Julie,
I'll let Gary answer your questions since he's the pro but you might want to rephrase using stair terminology. Here are the basics -
Boxed staircase - walls on both sides, with or without skirts.
Open staircase - wall on one side, railing on the other or rarely railing on both sides.
Stringer or carraige - usually 2x12 framing that supports the stairs.
Skirt - Baseboard like trim on the sides of the stairs. A boxed staircase would have 2 identical skirts, while an open staircase would have 1 skirt on the wall side above the treads and 1 mitered skirt on the open side, below the treads. Skirts are not structural - they are decorative.
Housed stringer - An inside skirt on an open staircase which is also structural (supports treads).
Tread - the horizontal stepping surface.
Riser - the vertical surface between treads.
Rise - the vertical rise of each step measured from tread to tread.
Run - the horizontal ditance of each tread measured from riser to riser.
I have Scott Schuttners book and it's the only decent book out there but be aware that his stair joinery is unecessarily complicated.
This book is normally only available if you buy parts from the manufacturer but I found an independent source and I'm waiting on it.
http://www.coffmanstairs.com/installer/installer5.htm
Also, it's always good to learn new things but I warn you that I have been doing basic carpentry (flooring, baseboard, crown molding, casing etc) for several years on my own houses and helping friends with theirs. I also have a full complement of shop tools - table saw, sliding miter saw, router table etc. I knew replacing my carpeted stairs would be difficult but that was understatement. It took me about 3 months just to get everything done except the railing working almost full time. Demolition of the existing stairs was extemely difficult and hard on my body and I sustained many hand injuries. Some of the existing framing cracked due to nail removal and had to be replaced. During much of this time I had the stairs w/o well attached treads and it was dangerous (I'm referring to an open staircase which is much more difficult than a boxed).
Building open staircases requires near furniture quality work built into the inexact framing of your house. It can be very frustrating to the uninitiated to say the least. You may want to consider solid flooring over existing treads. It's a fraction of the work and expertise level. The only real downside to this method IMO is the lack of a inside and/or outside skirt if one does not already exist. Of course it's not impossible to add skirts even when installing flooring over treads but I don't think it's common.