Others have already pointed out good enough references to begin with. I'm not listing a true Dummy's guide, but rather some pointers from the Unicode Consortium page. You'll find some more nitty-gritty reasons for the usage of different encodings at the Unicode Consortium pages.
The Unicode FAQ is a good enough place to answer some (not all) of your queries.
A more succinct answer on why Unicode exists, is present in the Newcomer's section of the Unicode website itself:
Unicode provides a unique number for
every character, no matter what the
platform, no matter what the program,
no matter what the language.
As far as the technical reasons for usage of UTF-8, UTF-16 or UTF-32 are concerned, the answer lies in the Technical Introduction to Unicode:
UTF-8 is popular for HTML and similar
protocols. UTF-8 is a way of
transforming all Unicode characters
into a variable length encoding of
bytes. It has the advantages that the
Unicode characters corresponding to
the familiar ASCII set have the same
byte values as ASCII, and that Unicode
characters transformed into UTF-8 can
be used with much existing software
without extensive software rewrites.
UTF-16 is popular in many environments
that need to balance efficient access
to characters with economical use of
storage. It is reasonably compact and
all the heavily used characters fit
into a single 16-bit code unit, while
all other characters are accessible
via pairs of 16-bit code units.
UTF-32 is popular where memory space
is no concern, but fixed width, single
code unit access to characters is
desired. Each Unicode character is
encoded in a single 32-bit code unit
when using UTF-32.
All three encoding forms need at most
4 bytes (or 32-bits) of data for each
character.
A general thumb rule is to use UTF-8 when the predominant languages supported by your application are spoken west of the Indus river, UTF-16 for the opposite (east of the Indus), and UTF-32 when you are concerned about utilizing characters with uniform storage.
By the way UTF-7 is not a Unicode standard and was designed primarily for use in mail applications.