Can native speakers of Mandarin Chinese readily understand speakers of neighbouring regions such as Korea, Japan, Mongolia, or Canton? Are Mandarin Chinese and any of these languages mutually intelligible?
No. Mandarin is part of the Sino-Tibetan language family, whereas Japanese, Mongolian, and Korean are not.
Even within the Sino-Tibetan family, most langauges are mutually unintelligble. Even worse, within the Chinese family, most regiolects and topolects are mutually unintelligble.
People in Shanghai, Suzhou, Ningbo speak Wu, which is not intelligble for Mandarin speaking people. The two topolects broke off some two thousand years ago.
People in Guangdong and Guangxi speak Cantonese, which is unintelligble for people speaking Wu or Mandarin. And so it continues. Only nearby topolects are intelligble, for instance Sichuan Mandarin vs. Hebei Mandarin vs. Tianjin Mandarin and so on.
Japanese has borrowed the Chinese writing system, but the spoken language has nothing in common with Chinese. Even so, Japanese has borrowed many loanwords, that often sound similar (电话 is denwa in Japanese, dianhua in Chinese). Chinese is therefore considered as the equivalent of Latin in East Asia, since many langauges have been influenced by Chinese.
At best, people from different nations can only pick up stray words in other languages.
On the other hand, they have long shared the same writing system, which means that Japanese can read Chinese and vice versa (when using characters, not Hiragana or Katakana), with some difficulty. This was true for Korean and Vietnamese before they incorporated their own writing systems.
It doesn't matter if the writing is in traditional or simplified, most can read both varieties, although may not be able to write in both scripts.
Adding to the great answer posted by @倪阔乐, one thing Westerners don't understand well is there are more than 200 mutually unintelligible languages in China, all spoken in the same country. According to the linked Wikipedia article on 'varieties of Chinese':
Chinese (汉语/漢語 Hànyǔ), also known as Sinitic, is a branch of the Sino-Tibetan language family consisting of hundreds of local language varieties, many of which are not mutually intelligible. The differences are similar to those within the Romance languages, with variation particularly strong in the more rugged southeast. These varieties, often called "dialects", have been classified into seven to ten groups, the largest being Mandarin (e.g. Beijing dialect), Wu (e.g. Shanghainese), Min (e.g. Taiwanese Hokkien), and Yue (e.g. Cantonese).
When you go to Shanghai and hear people there speaking Shanhainese, you will be surprised to find how different it is from Mandarine. Mandarine is often called Putonghwa (普通话, literally 'common speech') because there are many uncommon speeches in China. It was only 1930 when a standard national language was adopted in China and before then, most Chinese people spoke only their local languages.
As explained in the other answer, Chinese is Sino-Tibetan languages while Korean and Japanese are believed to be language isolate (there are some disputes over their origin, though). It is impossible they could be mutually intelligible.