A slight sampling of my somewhat mediocre knowledgs.........
"Road" Bike- Has more of an "aggressive" geometry. Current fashion is the compact frame with somewhat shorter top tub to allow for the aggressive riding position. Fork and head tube rake are as such to make for quicker, more responsive handling. New riders to Road bikes often refer to this as "twitchy" feeling. Seat tube angle is designed for lower posture. Modern road bikes are set up to typically have a much lower bar than seat relationship. Typically runs much more narrow (23-25) and higher pressure tires for less rolling Resistance. Wheels tend to be a lower spoke count these days.
Touring Bike - Much more relaxed geometry. less head tube and fork rake to allow for more stable handling. This is also assisted by the longer chain stays which also help eliminate heel strike on pannier bags. Typically, the handle bar and seat adjustment is such to allow for a more upright riding position. The tourer is usually heavier than the standard roadie. Runs a larger tire (32- 38) at lower pressure. Geared very low for loaded touring. Designed for long periods of saddle time.
Cross Bike - geometry wise is a combination of the two kinds. It is a bit more of a road geometry in the head and fork rake, and has less of an aggressive cockpit for the rider. It can typically run tires in line with the touring bikes and accommodates the knobbie approach as well. Used for off road racing, typically has cantilever or V brakes. Most models come with braze ons and eyelets for racks and fenders. If a rider maintains two sets of wheels, one with off road tires and one with road tires, you get the best of both worlds by merely changing wheels. The same can be accomplished with having a set of slicks and a set of knobbies, but your slicks will tend to be more wide, like a touring bike. Is much more rugged than a road bike, allowing for trail excursions and curb hopping, but less of a land yacht than a touring bike. Serves well as a commuter and light touring bike, but can do extended touring with a trailer.