In a horizontal style hive, the bees work the combs in - you guessed it - a horizontal fashion. Combs are built across the width or length. Brood is raised, usually, near the entrance and honey is stored farther back. It's internal volume is fixed, though some constructions offer limited expansion by various - often cumbersome - means. It requires a large amount of ground space but it completely stable in even the highest winds of during the roughest of handling. This style of hive is most prominent in equatorial regions of the world, but is also commonly found in the apiary of hobbyists just about everywhere.
Vertical hives are worked by the bees from the top, down. Honey is stored at the top of the combs and brood reared toward the bottom. The hive is often built to be easily expandable up, down or both by the simple addition of more boxes. These occupy a small amount of ground space but become unstable as they get taller, becoming prone to tipping in the wind or during handling. This style of hive dominates the world. It can be found from the equator to both poles.
Movable combs are attached to bars or within frames which can be readily re-located to other positions within the hive or placed into a different hive altogether. The two most common forms of moveable comb are bars across the top of the hive, with some means to guiding the comb to keep it straight, and frames - also with guides - to contain the comb. The only real difference between these two is ease of movement and method of harvest.
In a bar type, the comb is often attached to the side of the hive and these attachments must be cut in order to (re)move the comb. A wax or other guide is used to encourage the bees to build the comb straight across the bar and prevent attachment to other, adjacent, bars. In order to harvest, the comb is cut from the bars and then pressed to free the honey from the storage cells. Alternatively, the honey can be eaten while still in the comb. It is a very economical means of harvesting, requiring little equipment at low cost. Additionally, the bars are simple to construct with only the most basic of tools, little real skill, and use very few resources.
In a frame type, the combs are built inside a framework of wood or other material. There is no attachment to the box, itself, making it a simple matter to slide or lift the frame into its new location. A guide is still required to prevent the bees from building comb which wanders across more than one frame. Often, a sheet of wax or plastic with a hex pattern embossed is used give the bees a bit of a "head start" on their comb and to help regulate the size of cells built. To harvest liquid honey from the comb, a type of centrifuge is used to force the honey from the cells, after cutting the wax seals from them. These machines are expensive, require a large storage area when not in use (most of the year) and need routine maintenance to keep them working properly. Alternatively, the comb can be cut from the frame and processed in much the same way as bars or fixed comb.
The beekeeper - that's you - needs access to the interior of the hive. Otherwise, it's just a pretty lawn ornament with bees inside. You probably want to be able to get honey out of it, and maybe you even want to work on an intimate level with the bees and influence their breeding and other habits.
The most common arrangement is for the beekeeper to get in from the top by removing some sort of lid. This usually exposes the tops of the combs or comb-bearing frames, allowing the beekeeper to make whatever adjustments he deems necessary. If the combs are fixed, this only allows for honey harvest and usually is pretty messy and destructive. Movable combs or frames allow for much more finesse in your manipulations.
The second most common arrangement is side- or rear-access. This access direction exposes the height of the combs to the beekeeper and allows quite a lot non-destructive intrusion even in fixed-comb hives. In fixed-comb hives, however, the beekeeper is still largely relegated to harvesting the honey which has, hopefully, been deposited quite close to his access point.
Hives accessed from underneath have virtually disappeared. Skeps and a few other hive types require that they be tipped up and worked from underneath. This causes surprisingly little disturbance to the bees, even during extremely invasive procedures. These bottom-access hives are almost exclusively fixed comb arrangements, but there have been some rather arcane exceptions.
Clay can be found almost everywhere in various forms. It is low cost. It is easily worked with simple tools. It can be easily made into any thickness. Once dried and baked, it is relatively moisture resistant. It has very high impact resistance, making it unlikely to break from short falls. It has the unique characteristic of providing insulation from high heat and re-radiating it when outside temperatures cool, making it ideal for desert environs of high daytime temperatures and cold nights. It's major drawback is its weight, making it very difficult to move. This is a fantastic material for stationary apiaries or in areas where beehive theft is common.
Concrete shares most of the characteristics of clay with the added benefits of no requirement for baking and it can poured into molds for rapid hive construction.
Glass is expensive, heavy, difficult to cut and join, prone to breaking from even minor impacts and provides little insulating value. Its smooth surface is difficult for bees to attach comb to (same with glazing on clay hives). This can be an advantage for the sides of a hive, but is a clear disadvantage if considering its use for the top of a fixed-comb hive. Its transparency makes it ideal for observing the activities of the bees within the hive. However, this same transparency amplifies the heating effects of the sun and so the hive should be covered or kept in constant shade except for brief periods.
Wood is the dominant hive material around the world. It's outstanding combination of weight, sturdiness, ease of construction and weather tolerance makes it a near-perfect material for the purpose. Most of the areas where it does not dominate are poor countries with a small supply and, therefore, high price tag. In Europe and parts of south east Asia, this is being slowly overtaken by synthetic materials due to superior durability, weathering and similarity of cost.
Straw, once the king of hive-body material, has taken a back seat to other materials. It is low cost, light weight, easy to work, has a high insulating value and is resistant to decay. However, it provides poor protection from rain and is not very structurally sound, making it chancy to stack it high in vertically oriented hives. Also, modern harvesting combines break or chop the straws, making the availability of high quality long straws - essential to good construction - difficult to acquire. This is probably the primary reason for its reduction in use as a hive-body material. Its primary disadvantages are its many small cracks and crevices, which provide a safe hiding place for a variety of hive parasites, and its attractiveness to rodents.
Synthetic materials, most commonly high-density polyurethane and polypropylene (foamed or vacuum-formed), are continuing to fall in price and, so, are gaining popularity in parts of the world where beekeepers can afford to buy their equipment. The weight and price tag [of finished hive components] is comparable to wood. It is virtually immune to the effects of weather when coated with a UV protectant. It can easily be disinfected or cleaned with high pressure steam. It is sturdy and provides excellent insulation. It can be formed into any shape and many hive accessories made from this material are already in production. However, it cannot be made at home and is available in a limited variety of shapes and sizes. Its detractors claim its biggest drawback is its high environmental cost since the making of these plastics requires oil and other chemicals. This is a material which will be interesting to watch over the coming decades.
Standardized hive equipment reduces the beekeeper's overall financial costs in their apiary. The down-side is the potential un-suitability of this equipment for the individual beekeeper's needs. Migratory beekeepers have different needs than stationery apiaries and some beekeeper's have physical ailments which reduce the appropriateness of some hive types.