In the Chinese language, China is referred to as 中国 (zhōng guó), or 中國 in traditional script, meaning the middle kingdom. This wasn't always the case, as up until the Qing (清) dynasty, the country was referred to by the name of the ruling dynasty such as Han (汉) or Tang (唐). However, starting in the 17th century, China entered into formal agreements with other states and the term Middle Kingdom came to mean the country.
What China is referred to in other languages, however, is another matter. The ancient Greeks and Romans knew of Sēres (Σῆρες), the "land where silk comes from." The name is thought to derive from the Chinese word for silk, 丝 (sī), and is itself at the origin of the Latin for silk, sērica. Then there is the name Sīnae, which gave rise to the prefix "sino". The Arabs used the word Ṣīn صين, while the Persian term was Chīnī چین, itself derived most likely from Sanskrit. The traditional etymology is that all these terms trace their roots back to the Qin (秦) dynasty which was the first to unify China in the 3rd century BC.
What does all this have to do with the Portuguese? Well, the modern word "China" was first used by the Portuguese explorers of the 16th century, probably borrowed from the Persian. It was first recorded in 1516 in the journal of the Portuguese explorer Duarte Barbosa, which was later translated and published in England in 1555. More importantly, it was the Portuguese Jesuit missionary Bento de Góis who proved that the Cathay described by Marco Polo was in fact China. The word Cathay originates from the Manchurian tribe known as the Khitan that dominated Central Asia in the 12th century, and to this day the Kazakhs, Turkmen, Russians and even Ukrainians refer to China as Қытай or Китай.
Bento de Góis was the first known European to travel overland from India to China, via current day Afghanistan and the Pamir Mountains in the early 1600s. His is a story worth looking into.