"What size brood boxes and honey boxes (supers) would you like?"
When you buy your first hive, this will likely be the first question you're asked. Many would-be beekeepers reply, "What's the difference?" This early choice you make for your equipment really is a question of how you feel about weight and standardization.
These boxes are where your bees live and produce honey. The stack of boxes we call a hive is only walls, with a bottom board at the very bottom, and covers at the very top. The interior volume changes from a squat rectangle in spring to a tall rectangle during the summer, and returns to a rather short size for winter, as the beekeeper adds or removes boxes.
Remember that there's nothing special about any boxes as far as the bees are concerned. They'll live in a box of just about any shape or size. Your choice is really mostly about what works best for your beekeeping needs, and the bees will accept whatever you choose.
In Year One, your bees might only need brood boxes and may not build enough comb to start filling honey supers. Brood is the term for young bees, ranging from the eggs the queen laid up through the baby bees chewing out of their hexagonal cells as adult bees. Later on - maybe during your first summer or else in spring of your second year - the honey storage boxes ("supers") need to be ready to put on the hive too, but colonies grow best if allowed to fill their brood boxes first before more space is added.
The usual beginner hive kit consists of two "deep" (9-1/2 inches high) brood boxes and some number of "medium" (6 inches high) honey supers. In its second year of life, a strong colony might need 4 medium supers. But just because certain box sizes come in the standard beginner's kit, that doesn't mean those box sizes are right for you!
The chief reason to choose one box size over another is their weight once the bees have filled them. Brood is light compared to honey, and the bees arrange their home with brood below food (pollen and honey). For most of the season, the bottom box will typically contain only brood, and the top super will contain only honey or nectar. The topmost brood box will often contain a mix of brood, honey, and pollen. Thus, the highest boxes usually hold the heaviest stuff, with the contents getting lighter as you move down the hive.
More and more beekeepers are opting to exclusively use medium-depth boxes. The volume of the brood chamber and square inches of comb inside 3 mediums is just a tad larger than the space in 2 deeps, but it's so close that we describe 3 mediums and 2 deeps as equivalent. In this option, you'd use 3 mediums for brood, and then 3-4 more medium supers for the honey. Our Head Beekeeper Anne has always been grateful to the person who started her out, for steering her towards the simplicity of an all-medium set-up.
Using nothing but deep boxes is a choice often made by commercial beekeepers, giving the benefit of standardized box and frame sizes, and also reducing the total number of hive parts needed for any one hive. They can also get away with this because a commercial beekeeper's hired help is usually young, strong, and replaceable. The Seeley honey bee research lab at Cornell also used all-deep hives, for similar reasons - tired grad students graduate and are always replaced by young new ones who are willing to lift deep honey supers!
There are plenty of other variations you can consider. Some beekeepers use only a single deep as a brood chamber, while some use a medium and a deep for the brood chamber. These are possibilities you should leave until you've mastered the basics, with either two deeps or three mediums to provide your bees with a relatively large brood area and enough room to accommodate an overcrowded hive in swarming season.
There are plenty of ways to stack your boxes. If you wanted, you could use a pile of shallows as your brood boxes, and then put one or two giant deep-sized honey supers on top of them. You shouldn't, but you could. It's your apiary, your back, and your sanity. The bees will adapt to many different configurations if you offer them (though even bees have their limits). If you're getting started with bees, think carefully about which of these options best suits you.
Check out this Betterbee YouTube Video that covers these topics: Choosing Your Boxes.
Of course, once you choose your box depth, you also have to choose how many frames you'd like per box. Check out Choosing Equipment: 8 Frame vs. 10 Frame Boxes!