UTF-8 has an advantage in the case where ASCII characters represent the majority of characters in a block of text, because UTF-8 encodes these into 8 bits (like ASCII). It is also advantageous in that a UTF-8 file containing only ASCII characters has the same encoding as an ASCII file.
UTF-16 is better where ASCII is not predominant, since it uses 2 bytes per character, primarily. UTF-8 will start to use 3 or more bytes for the higher order characters where UTF-16 remains at just 2 bytes for most characters.
UTF-32 will cover all possible characters in 4 bytes. This makes it pretty bloated. I can't think of any advantage to using it.
The term is ambiguous, but in my internationalization work, we typically avoided the term "multibyte character sets" to refer to Unicode-based encodings. Generally, we used the term only for legacy encoding schemes that had one or more bytes to define each character (excluding encodings that require only one byte per character).
Shift-jis, jis, euc-jp, euc-kr, along with Chinese encodings are typically included.
Most of the legacy encodings, with some exceptions, require a sort of state machine model (or, more simply, a page swapping model) to process, and moving backwards in a text stream is complicated and error-prone. UTF-8 and UTF-16 do not suffer from this problem, as UTF-8 can be tested with a bitmask and UTF-16 can be tested against a range of surrogate pairs, so moving backward and forward in a non-pathological document can be done safely without major complexity.
A few legacy encodings, for languages like Thai and Vietnamese, have some of the complexity of multibyte character sets but are really just built on combining characters, and aren't generally lumped in with the broad term "multibyte."
Best Answer
Check out Joel Spolsky's The Absolute Minimum Every Software Developer Absolutely, Positively Must Know About Unicode and Character Sets (No Excuses!)
EDIT 20140523: Also, watch Characters, Symbols and the Unicode Miracle by Tom Scott on YouTube - it's just under ten minutes, and a wonderful explanation of the brilliant 'hack' that is UTF-8