The good news when it comes to choosing a hive is that you have a ton of options. The bad news... is that you have a ton of options! Like anything that has evolved over many years, the equipment and the approaches associated with beekeeping have evolved in many directions. With this in mind, one could almost fill a book describing all those options.
So, we will describe the common elements of the majority of hives. For each of the components below you do, indeed, have many choices but the basic principles are essentially the same.
We are going to build our standard hive from the ground up.
The bottom of the hive is the bottom board. While some bottom boards are solid, screened boards - incorporating a screen open to the space below - help with ventilation, which helps the colony keep the temperature at the right level, as well as removing moisture.
Often the bottom board (and therefore the rest of the hive) sits on a stand that raises the hive off the floor, to make working on the hive easier.
Often an "entrance reducer" is placed between the bottom board and the brood box (see below). The intent of this part is to, well, reduce the entrance (it's well named, eh?) to keep out unwanted creatures like mice. They are often used while a hive is establishing itself and it is common for them to be packaged with the bottom board.
The brood box sits on the bottom board and is, in many ways, the soul of the hive. This is where the queen will live - the hive is constructed to ensure she can't leave the brood box - and therefore where she will lay her eggs and young bees are raised. The brood box is also used by the colony to store reserves of pollen and honey.
Of the major components of the hive (see supers, below) the brood box is the deepest and largest. It houses a number of vertically hanging "frames", often 8 or 10 of them. A common size for a "deep" brood box is approximately 20" x 16" x 9 1/2". For some hobbyists this is a large and heavy unit to handle and so some prefer to use a less tall brood box, with a height of 6 1/2".
The queen is constrained to the brood box by placing a framed grid on top, which has gaps sufficient to allow worker bees to pass through, but not queens or drones.
Above the queen excluder there are generally one or more "supers". These are areas in which the colony can expand (in the summer, a lack of space can lead to swarming, so it's important to ensure the colony has sufficient space). The supers, which are less tall than the brood box, is used by worker bees to store honey.
Many beekeepers have three supers in total, including the brood box, during the active season, allowing the colony to expand and for the storage of yet more honey. The height of the super chosen is something of a tradeoff between being able to store more honey vs. ease of handling. As a guideline, a fully laden brood box can weigh 80 lbs or more - not easy to move around. Therefore, the honey supers are less tall, to reduce their weight, although they can still exceed 30 lbs.
Another tradeoff is associated with the claiming of honey by the beekeeper. The number of frames in a deeper super is the same as a shallow super but obviously has a greater surface area (read: volume of honey). Therefore larger supers allow the retrieval of more honey, for the same amount of physical work by the beekeeper.
Topping off the hive is the inner cover, along with a telescoping cover. The inner cover is helpful in ensuring bees don't "glue" the top of the uppermost super to the roof. It also forms an air pocket which is helpful in regulating temperature.
The telescoping cover generally has ventilation holes, which are essential to keep the hive dry.