Example sentences
nǐ你xué学hàn汉yǔ语duō多jiǔ久le了？
How long have you studied Chinese?
hàn汉yǔ语bù不nán难。
Chinese isn't hard.
About this word
汉 is the name of the Han people, the majority ethnic group, who took the name from the Han dynasty. 语 means "language, speech". So 汉语 is literally "the Han language". Because 语 points at speech rather than writing, 汉语 is the natural word for the language Chinese people speak, and it is the word inside the exam name 汉语水平考试 (HSK). 中文 is built with 文 ("writing") and leans toward the language as something written, so you hear 中文课 and 中文网站. 普通话 is Mandarin as the official standard. All three can translate English "Chinese", but they carve up the territory differently.
The story behind the character
汉
汉 was the name of a river, the Han River. Then Liu Bang founded the Han dynasty, and 汉人, 汉字, 汉语. Han people, Han characters, Han language. All borrowed the name from that one river. So the 汉 in "Chinese" began as a tributary of the Yangtze.
语
语 is 言, speech, with 吾 giving the sound. 吾 means I, but here it only supplies the pronunciation; in Old Chinese 吾 and 语 sounded the same. Xu Shen wrote 語, 論也, to discuss, to talk something over. So 汉语 is the speech of the Han, and 语言 is the general word for language.