Japanese kanji mostly borrow or adapt from Chinese kanji. However, like it’s industry, cooking or other field, Japanese also use, naturalize, adopt, and give the Chinese kanji new pronunciation/reading. Hence, Japanese kanji has two pronunciation (or can be said as reading):
1. Kun-yomi (Japanese Reading) 
This reading is used with hiragana character. People name also mostly reading by this type.
水 pronounced みず ( mizu) means water.
山 pronounced やま (yama) means mountain.
2. Onyomi (Chinese Reading) 
水 pronounced すい (sui) also means water. In chinese language, this character pronounced Shuǐ means water
山 pronounced san or zen means mountain. In chinese language, this character pronounced Shān means mountain.
Here is common mistake, Mount Fuji in Japanese language is Fuji-san (富士山) , not Fuji-yama because it in a compound with another kanji. To learn types of kanji compound, you can check this page.
However, name of person usually pronounce with Onyomi although it is in a compound. For example, 山本 pronounced Yamamoto not Sanmoto or Zenmoto.
Nanori: for name of person 
Actually, Japanese language also has a special reading/pronunciation for name of person. It is called by Nanori (名乗り）, literally means “self-introduction” .
For example, infamous Hg polution in the past. City of 水俣 pronouced Minamata. It not follow Onyomi or Kunyomi in which shoul be pronounced Mizumata or Suimata.
Please correct if there are some mistake. After all, I’m just a foreign person in Japanese with limited of Japanese.