The name "French Horn" is pretty misleading, but it is the most common name for the instrument in the United States. Why is it that, to most of the world, it's just called "horn"?
The horn has a long history, but let's fast forward to the 16th century and the hunting horn. The hunting horn was widely used in both Germany and France for sport. With the development of opera in the mid-16th century, the hunting horn had its premier as an on-stage, diegetic* music source. This eventually led to the horn being used in the pit as a regular instrument, not just as a prop or an extension of the plot.
The horn continued to develop and be utilized as a serious musical instrument, leading to advancements in how they were made. By the 18th century, the Germans had introduced moveable slides and crooks that could alter the pitch of the instrument, allowing the horn to be used more frequently.
The 19th century saw another leap for the horn with the creation of pistons and valves. Ask any horn player, and they will tell you what a miracle this advancement was! The inclusion of valves in the design of the horn allowed for chromatic pitches to be accessed without the use of the right hand. The invention also eliminated the need for a player to own or carry around a bunch of keyed crooks. A real game changer for technical (as well as lyrical) playing!
German makers made these advancements that led to the modern horn possible. A majority of horn players around Europe played instruments that were designed and built in Germany. British horn players in the 19th and 20th centuries broke away from this tradition, utilizing horns made in France. This is where the name "French horn" originated. The British used this name to distinguish themselves from "other" horn players.
The term "French horn" is not widely used in the UK today, but it did travel across the pond and is still the most common term for the instrument in the US.
The International Horn Society officially set the name of the horn to "horn" in 1971, just one year after the institution's founding. One of the reasons that "French" is still so common here in the US is the adoption of the term "horn" by several other instruments. This was brought on by the rise of Jazz as a genre, where any wind instrument was often called "horn." As a young player, I (like many others) called it a "French Horn."
Now, I say, "Actually, it's German" anytime anyone asks!
*Diegetic: occurring within the context of the story and able to be heard by the characters.