A Chinese radical (Chinese: 部首; pinyin: bùshǒu; "section header") is the part of a Chinese character used to list the character in a Chinese dictionary.  

The radical is usually a semantic indicator: a piece that gives hints to the reader about what the whole character means. Later, the term was given a second meaning, the 部首 (Pinyin: bùshǒu, Japanese bushu, Korean busu). This means  "section header", which refers to the place where a character is listed in the dictionary. 

Radicals are typically simpler characters, or different variants of simpler characters, which combine with other radicals to make a more complex character. People can only make so many unique ideographs for the huge number of words and morphemes that make up a language. Combining already existing ideographs together to form a compound ideograph is a very convenient solution to make many more words. While there are thousands of Chinese characters (well over 45,000) that exist, it is said that nearly all characters used in daily life are made up of just 214 radicals. 

One example of simpler characters combining to form a more complex one is the complex character 明 míng meaning "bright" or "tomorrow", which is made up of the radicals 日 rì meaning "sun" or "day" and 月 yuè meaning "moon" or "month", both of which are simple characters that can exist by themselves, and the putting together of sun 日 and moon 月 to make one character show that either the sun and moon pass before tomorrow comes, or that they are the bright objects in the sky that shine light on the earth. An example of characters with radicals that don't exist by themselves is the complex character 休 xìu meaning "to stop", “to cease” or "to rest", which is made up of the radical 亻, which is a variant of the simple character 人 rén meaning "person" or "human", though the variant never exists as by itself must therefore attach itself to another character to form a complete character, and the radical 木 mǔ meaning "wood" or "tree", hinting what a person might want to do after he/she ceases working (like lying on a tree). 

A more complex is the character 媽 mā meaning “mother” (shown right), in which the left part 女 nǚ meaning "woman" or "female"  happens to be the semantic component, and also the section header under which dictionaries list the graph. However, not all section headers function in a particular character as the semantic component. Some are based on phonetics, or hint as the character's pronunciation, like the right part 馬 mǎ meaning "horse" of the character 媽. Although the meaning of the character 媽 has no clear, obvious connection to a horse, 馬 is included to help the reader remember its pronunciation. Other section headers are merely chosen for convenience of finding the character in a dictionary, and play neither a semantic nor a phonetic role, like 二 èr "two" in 亞 yà "second" or "inferior",亅 jué (no meaning) in 了 liào "past tense marker", and 一 yī "one" in 丁 dīng "fourth". These two meanings of “radical” are not synonymous, though they may coincide in a particular example, causing a lot of confusion. (For clarity, both semantic component and section header terms are used here.)

The equivalents in cuneiform and Egyptian hieroglyphics are called determinatives.

Chinese character components, whether semantic or phonetic in role, are the building blocks for all Chinese Hanzi as well as in the derived forms of Japanese Kanji, Korean Hanja, and Vietnamese Chữ nôm and Chữ nho.
Since the Kangxi dictionary of 1716 was indexed using the 214 
Zìhuì section headers, the standard list thereof has remained unchanged. Section headers are foundationally important for the organisation and use of Hanzi, Kanji, and Hanja dictionaries.

References

Chinese language