A radical (from Latin radix, meaning "root") is the semantic root (i.e., portion bearing the meaning) of an inflected European word. Early Western sinologists borrowed this term to refer to the semantic component(s) of Chinese character (Hanzi). Later, the term was also borrowed for a second meaning, the 部首 (Pinyin: bùshǒu, Japanese bushu, Korean busu), literally meaning "section header", under which a character is listed in the dictionary.
For example, in the character 媽 mā or "mother" (pictured right), the left part 女 (pronounced nǚ in Mandarin Chinese) 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, while others are merely chosen for convenience, and play neither a semantic nor a phonetic role. These two meanings of "radical" are not synonymous, though they may coincide in a particular example, causing a great deal 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.

Meaning #1 of "radical" — semantic element
Since Chinese is not an alphabetic language, it is difficult to arrange the characters in a predictable way in a dictionary. Xu Shen categorized all the characters in his dictionary using a system of 540 graphical elements that he called bùshǒu (部首) or section header, found in different characters and often reflecting some common semantic or phonetic characteristic. These section headers are most commonly termed "radical" in English, which is intentionally avoided here to disambiguate meanings #1 and 2 of the term.
The graphic elements chosen by Xŭ Shèn were sometimes a semantic and sometimes a phonetic element. In other cases they were neither. However, for each character only one element was chosen, and this is thus referred to as "the section header" for that character.
Over time, Chinese lexicographers continued to refine this system for indexing Chinese characters, in order to collect and document them. The list of section headers was later trimmed to 214 in the 1615 dictionary Zìhuì. The Kangxi dictionary of 1716 was indexed using the Zìhuì section headers, and they form the standard list of 214 section headers or radicals still used today. Although there is some variation in such lists - depending primarily on what secondary section headers are also indexed - the canonical 214 headers of the Kangxi dictionary still serve as the basis for most modern Chinese dictionaries. Mei Yingzuo's Zìhuì was also the first dictionary to order the characters for each section headers using stroke count - the "section-header-and-stroke-count" method still used in the vast majority of present-day Chinese dictionaries.
In modern times, all new characters have been composed, in whole or in part, of components which also serve as section headers on the Kangxi list. However, they may in some characters be distorted somewhat in order to meet the requirement that all Chinese characters fit into a unit square. Viewed exclusively as graphic elements without connection to sound or meaning, it is always possible to completely decompose any Chinese character into a set of fundamental elements, although this requires some 500 elements rather than the 214 Kangxi section headers used to index characters in more recent dictionaries. (See List of Kangxi radicals).

Meaning #2 of "radical" — dictionary section header
In the characters 姐, 媽, 她, 好, 姓 and 妾, each character has a common graphical element: 女. In 妾, it is somewhat deformed in order to make the whole character fit into a unit square, but it is in each case present. 女 is also an independent character (Pinyin: nǚ), signifying a woman or the concept of femininity. This meaning is in some respect reflected in most of the characters above, although the relationship is sometimes historical, culture-specific, or even simply remote and uncertain. The other part of each of the above characters is used either for its phonetic value, playing no part in the meaning of the character but indicating something about its pronunciation; or as an indicator of meaning which has in some way been modified by the addition of 女 or which is used to modify its meaning, as shown in the table below:
In the images below, the red part of the character is the Section Header.
姐 jiě "elder sister" → "woman" + phonetic element 且 qiě
好 hǎo "good" → "woman" + 子 zǐ "child" (a cultural cliché of a woman with a child being the symbol of goodness; the true etymology is otherwise and unclear)
媽 mā "mother" → "woman" + phonetic element 馬 mǎ
姓 xìng "surname" → "woman" + 生 shēng "birth" (possibly a reference to ancient matrilineal customs)
她 tā "she", "her" - a pronoun modeled on its homophone 他 (tā "he", "him") by replacing the semantic component and section header 亻 ("man").
妾 qiè "empress", "mistress", "concubine" → "woman" + 立 lì "standing".
These examples intentionally use the section header for its semantic value, but this is not always the case. Furthermore, in these examples it is easy to identify 女 as a graphic element separate from the rest of the character. In other cases, identifying the semantic root and section header is not so simple.

Identifying Semantic Elements versus Section Headers in Characters
Note: The section below uses Unicode characters from the Kangxi Radicals block. These characters are not always available in common fonts.
In the examples above, five of the six characters have the section header on the left side (姐, 媽, 她, 好 and 姓) but it appears at the bottom in 妾. There is no fixed rule about where it can go in a character - it may appear in any position in a character. However, there is one pair of section headers that have the same shape, but are indexed as different section headers depending on where they appear in the character: 阝 (the abbreviated section header form of 邑, Pinyin: yì; see below) as in 都, is always on the right side of characters, while 阝 (the abbreviated section header form of 阜, Pinyin fù) as in 陽, is always on the left.
In writing, many components are distorted or change in form in order to fit into a block with other components. They may be narrowed, shortened, or may have different shapes entirely. Changes in shape, rather than simple distortion, may result in a reduction in the number of strokes used to write it. In some cases, these written forms may have several variants. The actual shape of the component when it is used in a character can depend on its placement with respect to the other elements in the character. In the image to the right, the colour blue is used for "irregular" forms.
Some of the most important variant written forms (except for ⻏ → 邑 and ⻖ → 阜 which have already been discussed):
(*) 心 becomes ⺗ when written at the bottom of a character.
In the above examples, it makes no difference if the component is used for its meaning or not. The changes in form remain the same.
The character simplification adopted in the People's Republic of China and elsewhere has modified a number of components. This has created a number of new section header forms: 食 is written 飠 when it forms a part of other traditional characters, but is written 饣 in simplified characters.

刀 "knife" → 刂 when placed to the right of other elements:

• examples: 分, 召 ~ 刖
  counter-example: 切
  人 "man" → 亻 on the left:
  
  
  
  • 囚, 仄, 坐 ~ 他
    心 "heart" → 忄 on the left:
    
    
    
    • 杺, 您, 恭* ~ 快
      手 "hand" → 扌 on the left:
      
      
      
      • 杽, 拏, 掱 ~ 扡,
        counter-example: 掰,
        水 "water" → 氵 on the left:
        
        
        
        • 汆, 呇, 沊 ~ 池,
          counter-example: 沝,
          火 "fire" → 灬 at the bottom:
          
          
          
          • 伙, 緋, 灱 ~ 黑,
            counter-example: 災,
            犬 "dog" → 犭 on the left:
            
            
            
            • 伏, 突, 嵇 ~ 狙,
              目 "eye" → rotated 90˚:
              
              
              
              • 助, 見, 盲 ~ 曼. Limitations of the section header system
                
                Learning and using semantic components and section headers
                Learning to write Chinese characters, even the most complicated, means mastering the fundamental graphic components of characters and then learning which ones are used in particular characters and how they are combined. This is much easier than learning to write each character as a whole through pure memorization. For example, one can remember how to write 義 (yì, "right conduct") by knowing that it consists of a 羊 (yáng , "sheep") above 我 (wǒ, "I", "me" — one of the first characters learned in Chinese classes, but not listed as a section header in dictionaries). The same applies to the very complicated 30-stroke character 鸞 (luán, a mythical bird from Chinese mythology): 糸, 言, 糸 over 鳥. (See the image to the left.) In this case, all four components are dictionary section headers. Even though there are in total some 50,000 Chinese characters (although a far smaller number — some 4,000 to 6,000 — are in daily use), all characters no matter how complex can be decomposed in this way.
                Learning to write characters based on graphical decomposition requires no knowledge of whether the component elements relate to the character's meaning or to its pronunciation. Once the 200-odd dictionary section headers have been mastered, the amount of memorization needed to master Chinese writing is dramatically diminished. This is somewhat like learning the alphabet in languages that use them: Once a relatively small set of letters have been mastered, the spelling of a word involves only memorizing the chain of letters used to write it. Although Chinese components are not comparable in function to alphabet letters, they play to some extent a similar role in Chinese writing.
                
                Character Decomposition
                Most dictionaries use section header classification to index and lookup characters, although many present-day dictionaries supplement it with other methods as well. Following the "section-header-and-stroke-count" method of Mei Yingzuo, characters are listed by their section header and then ordered by the number of strokes needed to write them.
                The steps involved in looking up a character are:
                For example, consider the character 信 xìn, meaning "truth", "faith", "sincerity", and "trust". Its section header is "human" (人) and there are 7 additional strokes in the remaining portion (言). To look this character up in a dictionary, one finds the section header for "human" in the part of dictionary that indexes section headers, finds the page for that section header, and then passes through the lists of characters with one additional stroke, 2 additional strokes, etc. until one reaches the entries with 7 additional strokes. If the section header chosen by the user matches the section header used by the dictionary compiler (which can be difficult to guarantee for more complicated characters), and if both the user and the dictionary compiler count strokes the same way (also often a problem with characters that the user is unfamiliar with), the entry will be in that list, and will appear next to an entry number or a page number where the full dictionary entry for that character can be found.
                As a rule of thumb, components in the left or top of the character, or elements which surround the rest of the character are the ones most likely to be used as section header. For example, 信 is typically indexed under the left-side component 人 instead of the right-side 言; and 套 is typically indexed under the top 大 instead of the bottom 長. There are, however, idiosyncratic differences between dictionaries, and except for simple cases, the same character cannot be assumed to be indexed the same way in two different dictionaries.
                In order to further ease dictionary lookup, dictionaries sometimes list section headers both under the number of strokes used to write their canonical form and under the number of strokes used to write their variant forms. For example, 心 can be listed as a four-stroke section header but might also be listed as a three-stroke section header because it is usually written as 忄 when it forms a part of another character. This means that the dictionary user need not know that the two are etymologically identical.
                It is sometimes possible to find a single character indexed under multiple radicals. For example, many dictionaries list 義 under either 羊 or 戈 (the section header of its lower part 我). Furthermore, with digital dictionaries, it is now possible to search for characters by cross-reference. Using this multi-component method (which you can try out at Jim Breen's WWWJDIC Server), a relatively new development enabled by computing technology, the user can select all of a character's components from a table and the computer will present a list of matching characters. This eliminates the guesswork of choosing the correct section header and calculating the correct stroke count, and cuts down searching time significantly. One can query for characters containing both 羊 and 戈, and get back only five characters (羢, 義, 儀, 羬 and 羲) to search through.
                
                Identify the section header under which the character is most likely to have been indexed.
                Find the section of the dictionary associated with that section header.
                Count the number of brush or pen strokes in the remaining portion of the character.
                Find the pages listing characters under that section header that have that number of additional strokes.
                Find the appropriate entry or experiment with different choices for steps 1 and 3. Variations in the number of section headers