The most common hive in North America and Australia, the Langstroth hive was invented in the mid 1800s by the reverend Lorenzo Langstroth. The most important part of his invention, and that which he is most known for, is the moveable frame. There is much debate as to whether he truly was the first to create the moveable frame, but it is his hive that rose up to dominate the modern beekeeping world.
The Langstroth hive is made up multiple stacked boxes of various heights. The different box names are: deep, medium and shallow, which corresponds to the height of each box (deeps being the tallest boxes and shallows being the shortest). Most Langstroth beekeepers start their hive with either 2 deeps, or 3 mediums depending on their preference.
The boxes come in 2 widths -- 10-frame (the most common) or 8-frame. In our own Langstroth hives we use 8-frame medium boxes throughout the hive. This dramatically reduces the weight of the boxes, and makes all parts interchangeable.
As the colony builds through the boxes, the beekeeper usually adds boxes to the top (called supering) so the bees can continue building upward. Later in the season, after the colony has filled up the "supers," the beekeeper will remove the surplus honey supers, leaving the rest for the bees.
Inside the frames, most beekeepers use a thin sheet called foundation. These sheets of foundation are made from either beeswax or plastic, and they are pressed into the hexagon pattern to resemble that of a real comb. Upon installation, the bees draw this foundation out further, creating the cells into which their brood will develop and their honey will be stored. Often, this foundation has wires run through it for added strength. When it's time to harvest, the beekeeper will usually remove the frames and foundation, cut the caps off of the honey cells, and then spin the honey out in a centrifuge called an extractor.
What many don't realize is that when Langstroth first invented his hive, foundation wasn't readily available. Instead, he allowed the bees to build natural combs inside the frames (as they've built for millions of years). This is what we do in all of our hives.