In the Middle East, everybody calls this country something more like "Urdun" or "Ordon". How did it come to be called "Jordan" in the West? What is the history of the name of this country?
This is because the English word for the country of Jordan is ultimately derived from the biblical Hebrew word for the Jordan River, not from the modern Arabic name for the country.
After World War I, the region was made a British protectorate called Transjordan (as Pieter Geerkens points out), although English use of "Transjordan" predates the protectorate. Transjordan roughly means "the other side of the Jordan River." This name obviously doesn't make much sense to the people actually living in Jordan, so after independence the country's common English name was shortened to "Jordan."
So with that political history aside, we need to explain where that "J" in the river's name came from. That answer recapitulates in miniature the historical process that created the oldest English translations of the Hebrew Bible, which were based on Latin translations of Greek translations of the original Hebrew.
The Jordan River is referenced in the Hebrew Bible as the "Yarden". It's very common for English versions of Hebrew words to use a "J" where the biblical Hebrew has a yud ("y" sound). Yehoshua became Joshua, Yahweh became Jehova, and the Yarden River became the Jordan River.
This has something to do with transliterating of words from Hebrew to Greek to Latin to English. That's three different alphabets and four different phonetic systems. Hebrew and English both have a letter for the "Y" phoneme, but Greek and Latin don't, so something got lost in transliteration.
We can see a similar process in the transition from the Hebrew "Yehuda" to the English "Judea." The Greeks represented the "Y" sound in writing with an IO combination, which was then represented in Latin by the IU combination. At some point, "J" was added to the Latin alphabet, and a lot of leading "I's" in Latin became written as J's.
English versions of these words were often borrowed directly from the written Latin versions. English speakers did not begin to consistently differentiate i and j until the 17th century, so they came to pronounce these words with the hard "J" that you find in the modern English pronunciation of Jordan--and Judea, Joshua, Jerusalem, Jehovah, Jehoshaphat, and so on--instead of the soft J that you find in words like "Hallelujah."