A brief History
Before the invention of the honey extractor around 1865 honey was either squeezed or pressed from the comb or produced as comb honey.
Rather than cut combs up, it was easier to entice bees to build comb in small conveniently sized boxes or "sections". Moses Quinby, who is generally thought to have been the first commercial beekeeper in the U.S.A., produced section honey in the 1830/40s but didn't claim the method originated with him. It seems they weren't the small frames we are familiar with and it was before movable comb hives, so presumably box hives, perhaps with a hole cut in the top. One of the first beekeepers given credit for introducing something similar to the modern section was J.B. Harbison in 1857 who moved from Pennsylvania to California. They appear to have always been called sections from the start in all countries.
Harbison's sections were four pieces of thin wood that were nailed together, but this was very time consuming. The answer was found by using one long strip of wood and making three "V" shaped grooves partly through the wood so it folded into a box. There were comb joints that allowed it to clip and hold together. I have come across three people who are credited with being amongst the first to do this, J. Lee in the U.K. and A.J.Cook in the U.S.A. both in 1876 and a U.S court case of 1888 between James Forncrook and Amos I.Root stating that Forncrook had filed for a patent in 1879 that was granted in 1881. You will note this is after both Lee and Cook.
It was found that lime (Tilia spp) or basswood as it is called in the U.S. was good for this purpose on account of the grain being long and fine.
Without some sort of guide or restraint bees rarely build good honey combs of a similar width, or without building bridges between them. This makes comb honey unsightly and variable in thickness. To overcome this G.T. Wheeler patented a tin separator in the U.S. in 1870. This was presumably what we know as the section divider. I recall once seeing reference to thin pieces of wood used as dividers, but that may simply have been conjecture.
Sections are notoriously difficult to produce. They need strong colonies and good nectar flows. They are therefore probably more successful where the climate is warm and reliable. The section rack is the standard method of producing sections, but in some countries the hanging section frame is used, so a smaller quantity may be attempted.
Although sections appear to have been used before the invention of the moveable comb hive, I suspect that was the trigger for the increased popularity. From drawings I have seen it is clear that sections were also used in conjunction with skeps. Perhaps this is not surprising as skeps were in common use well into the 20th century. In those days many beekeepers worked for low wages on the land and probably couldn't afford a honey extractor.
Round sections are currently very popular, but the first recorded use was in 1889 with an article by T.Bonner Chambers suggesting slicing up glass jars and allowing bees to build comb in them. The first plastic round section was introduced in 1954
With the development of plastics, other systems have been introduced including Bee-O-Pac and Hogg half-comb cassettes. A websearch will reveal others.
The above has been gleaned from a number of sources. Where I have found conflict, I have tried to use a bit of logic to determine the more likely account. If you are able to add to or correct the above please Email me.