What variables best determine bicycle fit for an individual without riding it? Trying to decide a bicycle purchase, not sure how to choose the best fit bike for my body type.  Should I measure my leg length, torso length, arm length, body build, does my gender make a difference?
One thing that does make a major difference is the type of bike.  As a very general rule, the faster the bike is intended to be ridden, the more important the fit.  I'll address this from the point of view of what most beginners ride -- things like Euro-style city bikes, hybrids, and basic mountain bikes.  

So most bikes that most beginners (based on your tags) would start riding on are quite forgiving. For a conventional geometry, leg length (i.e. seat tube length) is the biggest factor.  This is the key number in the frame size, and the frame is the most important thing to get right as there are no (cost-effective) adjustments or substitutions possible.  When cheap bikes are sized by a single number, it's usually based on this length, but comparison between makes isn't simply a matter of the length.  To some extent a too-small frame can be compensated for by raising the saddle, but it's much better to keep this for fine-tuning as too much seat post extension often results in the handlebars being too low. A frame that's too big can lead to disomfort or injury, even if you don't have to worry about a top tube.

The reach (approximately top tube+ stem length) and stack (handlebar height) are secondary.  They scale with the seat tube so should be about right for a typically-proportioned person. Both are also quite adjustable, though to get the most adjustment means swapping out parts.

The effect of this is that most people can buy a flat-bar bike based on top-tube height / frame size alone, and only worry about anything more than the most basic adjustments when they start considering longer rides.  Of course extra care is needed if you have back/leg etc. problems.