Other than the geometry and frame materials used for a bike, the principle differences between them will be the type of gears and brakes they have. These systems will influence things like cost, weight, amount of maintenance, and efficiency of your ride.
The following sections may help you narrow your bike options:
Derailleur gears push (or derail) the chain from one sprocket (at the back) and one chainring (at the front) to another. Changing them changes the gear ratio, or how far the bike travels per crank revolution. Some bikes have as many as 11 sprockets and 3 chainrings, for 33 gear choices. Derailleur gearing offers the greatest possible choice of gears, but the pieces are all exposed to the elements, and require regular cleaning and lubricating to keep them working properly. As well, the rear derailleur can be damaged if your bike falls over.
Hub gears have a series of cog sets tucked inside the rear hub and a single sprocket on the outside. Moving the system's central cog from one cog set to another, changes the gear ratio. A control on the handlebar attached to a cable pulls the cog, and an internal spring pushes it back. Since the chain doesn't switch chainrings or sprockets, pedaling is not required to change gears.
The cogs aren't exposed to dirt and water, and require very little maintenance, other than an overhaul every 5-10 years. (The exposed chain and sprocket should still be cleaned and oiled regularly.) However, cogs take up more space than sprockets, so fewer gear choices can squeeze inside the hub (typically 8 to 10), and the total gear range is less than derailleur systems.
Rim brakes aren't as powerful as disc brakes, but they work in similar ways: a caliper pushes pads against the rim and friction slows the bike. Because your bike rims sit a couple of inches above the road, they get wet, which reduces friction and braking efficiency. Rim systems (sidepull, centrepull, cantilever, and V-brake) all have calipers that are relatively flexible, so energy that could press the brake pads against the rim is lost to caliper flex.
Disc brakes are usually made from thin steel plate that can withstand high crushing forces from the brake caliper. Discs are small diameter and positioned close to the hub, so they are less affected by dirt and water than rims. The calipers are very rigid, and less energy is lost to flex. The pad materials is more durable than that used for rim brakes, and stops well even if overheated on a long downhill.
Mechanical disc brakes are more powerful than rim brakes, but less powerful than hydraulic brakes. Mechanical brakes use wire cables within a housing to transfer force to the caliper. Some force is absorbed by stretch in the cable, some by compression in the cable housing, and some by friction between the cable and housing. These losses reduce the amount of force available to press the pads against the disc.
Hydraulic disc brakes use fluid within a sealed line to transfer force to the caliper. Hydraulic fluid doesn't compress, and loses very little energy to friction as it moves within its brake line. The pads can be pressed very firmly against the disc, which increases stopping power. The cables and housings on mechanical brakes must be lubricated or replaced from time to time, but they are easy for a home mechanic to service, and are available at any bike store. Over time, hydraulic fluid will get contaminated. Replacement may be too complicated for the home mechanic, and fluid and lines may not be available everywhere.
Read about Choosing Your Frame Size
Learn about Frame Geometry