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In 1999 I spent some months converting books in the Dewey Decimal classification to the National Library of Medicine classification in two medical libraries. I did not actually concern myself with Dewey, doing the classifying from each book's contents. However, I knew something about the Dewey Decimal system and here is a summary.

Dewey Decimal Classification
The Dewey Decimal Classification was invented in 1873 by Melvil Dewey and published in 1876, being expanded in subsequent publications. In 2002 it was up to the 21s. It is probably the most common system, followed by the Library of Congress system (which also uses Dewey!).
The Dewey Decimal Classification (DDC, or "Dewey") operates in positions of tens, with the first number classes, the second divisions and the third sections. Then comes a decimal point, and divisions by ten continue.
The hierarchy of classification is by discipline for which the work is intended, not by topic (or subject). For example, electronics can be within science or within technology depending on its application. So the classifications are:
 
000 Generalities (Encyclopedias, Newspapers, Periodicals, Information, Knowledge, Libraries, Computers, Journalism, Bibliography
100 Philosophy, Parapsychology and Occultism, Psychology
200 Religion
300 Social Sciences
400 Language
500 Natural Sciences and Mathematics
600 Technology (Applied Sciences)
700 The Arts Fine and decorative arts
800 Literature (BellesLettres) and Rhetoric
900 Geography, History, and auxiliary disciplines
 
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So the numbers start with 0 and run to 9. There is no limit to specificity after the decimal point.
The numbers are hierarchical. Hierarchy in the DDC is intentionally expressed through both notation and structure.
Notation: numbers at any one level are subordinate to a class one digit shorter, equal with a class with the same number of significant digits and superordinate to a class with longer numbers. The fewer the numbers the more general and all encompassing the topic, the longer the numbers the more specific. Where topics violate this then notes are given.
Structure: every topic is within all the broader topics above it, rather like an inverted tree, up to the highest ten disciplines. What is true of the whole is true of the parts. Any note for a class runs right down all subordinate classes. Therefore a note only needs to be given once at the highest level relevant.
Obviously the categories reflect real world experience. Once the numbers are set, however, it is a case of fitting text sources to them by taking the whole, majority or emphasis of an applied subject and then going on subsequent details.
Arguably giving one Dewey number to a source ends up more like an art than a science. Material in one source comparable to another may nevertheless have a completely different Dewey number because of the assessed emphasis and application of the whole. Web pages can be classified according to Dewey but they change and are often highly variable within.
Classifying a work properly uses these sources:
  • The title, but this can be misleading
  • The table of contents, if existing and adequately descriptive
  • Chapter headings and subheadings
  • Introductions that show the subject of the work and its place in thought
  • A preface (or author's introduction) giving purposes and afterthoughts
  • Bibliography, footnotes and references
  • Existing catalogue data
  • The text
  • Reviews
  • Further reference tools
  • Experts
Incidentally, works of the imagination are usually classified by literary form instead of topic.
So once the topic/s being contributed to by the author are found, the classifier goes to the schedules and its hierarchy to find the correct numbering.
With two or more topics then the classification goes with:
  • Where the topic is being applied, known as the "rule of application"
  • The fuller topic of the group
  • The lower number in the schedule (where there is equal treatment between two) unless instructed except where topics are major subdivisions and so go higher and more general
  • The higher and more general number with three or more equal topics
These work in cases of one discipline, but sometimes sources cross over disciplines:
  • Interdisciplinary numbers within disciplines down the hierarchy for certain topic areas
  • One discipline may be more important than another (the first digit alters)
  • With no interdisciplinary number the discipline that dominates gets the classification
  • A possibility for classification is within 000
  • After this it is like with more than one subject
Even then there may be several classification probablities. There are further guidelines:
  • Avoid zeros at the same hierarchical point keep as low as possible.
  • Use the Table of last resort with caution in this order:
  • Kinds of things
  • Parts of things
  • Materials by which things, kinds, or parts are made
  • Their properties
  • The Processes used that maintain the properties
  • Operations upon things, kinds, parts, or materials
An important point throughout is the author's intentions in applying his topic/s. Where classification flouts these, the classification cannot be right.

Dewey Cutter
This is where it gets a bit more complicated, as every source really needs a unique number (though a lot of libraries don't bother). The Dewey Cutter is a code number which, shown after a space or on the next line, keeps authors' names in alphabetical order. There are two versions - the Cutter-Sanbourn Four-Figure table and the Cutter Four-Figure table.
For example Worsfold, Adrian is W897 in the Cutter Four-Figure table and W931 in the Cutter-Sanborn Four-Figure table. There are tables and download programs to show these.
The first letter/s of book titles can be added to the Cutter number when the author has several titles. Literary criticism of the author means a Z is added to the Cutter number. In biographies, nothing is added there but the biographer's Cutter number comes a space or next line after the biographee's number.
Multiple copies get an added copy number. Updates get a year added. if there is a special location, they form a letter above the whole Dewy number (eg oversize area).

Criticisms
These must be the artificiality of applying one number to a work even though the idea of application not pure subject is sound. More than this is the intention of notational hierarchy. The same length of number does not indicate parity between entities. As Dewey has faced additions, it has become messy compared with the ways of the world, so that for example in contemporary times the Internet is condemed to have an awkward number (among many) like 004.678. Search engines just trawl through text! It also has a Western and American bias. Its coverage of religion is awful. Judaism's Torah is put within a Christian category of Old Testament, popular psychology and the religious now New Age/ Witchcraft got muddled in the 130's, and Buddhism has 294.3 compared with 288 for the small Unitarian denomination (arguably no longer Christian, so this is wrong, and its equivalent Society of Friends has 289.6). The Bahai religion is placed within Islam, which is should not have been, and the Shinto religion has a ridiculous 299.561. Dewey is like a Ptolomaic system long after Copernicus has realised the world works differently. Dewey is like the person who asks directions only to be told s/he shouldn't start from here.