Transcription is the conversion of a text from another medium. 

That can be the conversion of human speech into written, typewritten or printed form. But it can also mean the scanning of books and making digital versions. A transcriber is a person who performs transcriptions.

Transcription as going from sound to script must be distinguished from transliteration, which creates a mapping from one script to another that is designed to match the original script as directly as possible. 

Standard transcription schemes for linguistic purposes include the International Phonetic Alphabet (IPA), and its ASCII equivalent, SAMPA. See also phonetic transcription

In this table IPA is an example of phonetic transcription of the name of the former Russian president known in English as Boris Yeltsin, followed by accepted hybrid forms in various languages. Note that 'Boris' is a transliteration rather than transcription in strict sense. 

The same words are likely to be transcribed differently under different systems. For example, the Mandarin Chinese name for the capital of China is Beijing in the commonly-used contemporary system Hanyu Pinyin, and in the historically significant Wade Giles system, it is written Pei-Ching.

Practical transcription can be done into a non-alphabetic language too. For example, in a Hong Kong Newspaper, George Bush's name is transliterated into two Chinese characters that sounds like "Bou-sū" (布殊) by using the characters that mean "cloth" and "special". Similarly, many words from English and other Western European languages are borrowed in Japanese and are transcribed using Katakana, one of the Japanese syllabaries.

After transcribing 

After transcribing a word from one language to the script of another language:
 one or both languages may develop further. The original correspondence between the sounds of the two languages may change, and so the pronunciation of the transcribed word develops in a different direction than the original pronunciation.
 the transcribed word may be adopted as a loan word in another language with the same script. This often leads to a different pronunciation and spelling than a direct transcription.

This is especially evident for Greek loan words and proper names. Greek words are normally first transcribed to Latin (according to their old pronunciations), and then loaned into other languages, and finally the loan word has developed according to the rules of the goal language. For example, Aristotle is the currently used English form of the name of the philosopher
whose name in Greek is spelled  (Aristotélēs),
which was transcribed to Latin Aristóteles, from where it was loaned into other languages
and followed their linguistic development.(In "classical" Greek of Aristotle's time, lower-case letters were not used, and the name was spelled .)

When this process continues over several languages, it may fail miserably in conveying the original pronunciation.
One ancient example is the Sanskrit word dhyāna which transcribed into the Chinese word Ch'an through Buddhist scriptures. Ch'an (禪 Zen Buddhism) was transcribed from Japanese (ゼン zen) to Zen in English. dhyāna to Zen is quite a change.

Another complex problem is the subsequent change in "preferred" transcription.
For instance, the word describing a philosophy or religion in China was popularized in English as Tao and given the termination -ism to produce an English word Taoism.
That transcription reflects the Wade-Giles system.
More recent Pinyin transliterations produce Dao and Daoism.

Footnotes

Related pages

 Romanization
 Transliteration

Other websites 
 Transcripts of software freedom talks, with reasoning for the usefulness of transcripts 

Writing