In an earlier post, I explained how China became known as China thanks to the Portuguese: https://www.portuguese.asia/post/how-china-became-known-as-china-thanks-to-the-portuguese
As for what the Western world calls Standard Chinese––Mandarin––I always thought this came from the term 满大人 (mǎndàrén), which roughly translates as "His Manchurian Excellency". This made sense as the Qing dynasty ruled China from 1644 until imperial rule was overthrown in the 1911 revolution, a period when China came into growing contact with Europe. Qing rule was established by the Manchurians after they invaded Ming China from what is today northeast China, formerly referred to as Manchuria.
To my surprise, I just learned that this is apocryphal. The term Mandarin has been used by Europeans to refer to Chinese bureaucrats, chosen through the imperial examination for scholars, since the Ming dynasty. And the first Europeans to reach China in the modern era, during the Ming dynasty, were... the Portuguese.
After conquering Malacca in 1511, the Portuguese king Manuel I sent an embassy to the Chinese Emperor Zhengde (正德) in 1516. Led by Lisbon-born apothecary Tomé Pires, the embassy did not manage to be granted an imperial audience in part because Malacca was considered a Chinese protectorate, and the deposed Sultan of Malacca had promptly complained to the Chinese emperor. Certain members of Tomé Pires's delegation ended up being prisoners, and it was in the letters they sent that the use of the word mandarin (as it was spelled in Old Portuguese) was first recorded to refer to Chinese officials.
Now where did this word come from? There are some who think that it is derived from the Portuguese mandador ("one who commands") and mandar ("to command"). However, the general consensus is that the Portuguese were using a variation of the Malay word for government minister, menteri, which they picked up in Malacca. This came originally from the Sanskrit word mantri ("person who thinks and says") and versions of this word are still used to mean minister not only in India and Bangladesh, but also in Thailand, Cambodia, Malaysia and Indonesia.
Now what do the Chinese themselves call the standard version of their language in Chinese? Confusingly, this depends on geography. In mainland China, it's 普通话 (pǔtōnghuà) meaning common or plain speech, while in Taiwan it's 國語 (guóyǔ) or national language. In Singapore and Malaysia where the national language is Malay, the term used is 华语 (huáyǔ), which uses the word 华 to refer to Chinese cultural ancestry.
In English and most European languages, it's Mandarin Chinese, thanks to the Portuguese!