To expand on the answers others have given:
We've got lots of languages with lots of characters that computers should ideally display. Unicode assigns each character a unique number, or code point.
Computers deal with such numbers as bytes... skipping a bit of history here and ignoring memory addressing issues, 8-bit computers would treat an 8-bit byte as the largest numerical unit easily represented on the hardware, 16-bit computers would expand that to two bytes, and so forth.
Old character encodings such as ASCII are from the (pre-) 8-bit era, and try to cram the dominant language in computing at the time, i.e. English, into numbers ranging from 0 to 127 (7 bits). With 26 letters in the alphabet, both in capital and non-capital form, numbers and punctuation signs, that worked pretty well. ASCII got extended by an 8th bit for other, non-English languages, but the additional 128 numbers/code points made available by this expansion would be mapped to different characters depending on the language being displayed. The ISO-8859 standards are the most common forms of this mapping; ISO-8859-1 and ISO-8859-15 (also known as ISO-Latin-1, latin1, and yes there are two different versions of the 8859 ISO standard as well).
But that's not enough when you want to represent characters from more than one language, so cramming all available characters into a single byte just won't work.
There are essentially two different types of encodings: one expands the value range by adding more bits. Examples of these encodings would be UCS2 (2 bytes = 16 bits) and UCS4 (4 bytes = 32 bits). They suffer from inherently the same problem as the ASCII and ISO-8859 standards, as their value range is still limited, even if the limit is vastly higher.
The other type of encoding uses a variable number of bytes per character, and the most commonly known encodings for this are the UTF encodings. All UTF encodings work in roughly the same manner: you choose a unit size, which for UTF-8 is 8 bits, for UTF-16 is 16 bits, and for UTF-32 is 32 bits. The standard then defines a few of these bits as flags: if they're set, then the next unit in a sequence of units is to be considered part of the same character. If they're not set, this unit represents one character fully. Thus the most common (English) characters only occupy one byte in UTF-8 (two in UTF-16, 4 in UTF-32), but other language characters can occupy six bytes or more.
Multi-byte encodings (I should say multi-unit after the above explanation) have the advantage that they are relatively space-efficient, but the downside that operations such as finding substrings, comparisons, etc. all have to decode the characters to unicode code points before such operations can be performed (there are some shortcuts, though).
Both the UCS standards and the UTF standards encode the code points as defined in Unicode. In theory, those encodings could be used to encode any number (within the range the encoding supports) - but of course these encodings were made to encode Unicode code points. And that's your relationship between them.
Windows handles so-called "Unicode" strings as UTF-16 strings, while most UNIXes default to UTF-8 these days. Communications protocols such as HTTP tend to work best with UTF-8, as the unit size in UTF-8 is the same as in ASCII, and most such protocols were designed in the ASCII era. On the other hand, UTF-16 gives the best average space/processing performance when representing all living languages.
The Unicode standard defines fewer code points than can be represented in 32 bits. Thus for all practical purposes, UTF-32 and UCS4 became the same encoding, as you're unlikely to have to deal with multi-unit characters in UTF-32.
Hope that fills in some details.
A ‘character set’ is just what it says: a properly-specified list of distinct characters.
An ‘encoding’ is a mapping between a character set (typically Unicode today) and a (usually byte-based) technical representation of the characters.
UTF-8 is an encoding, but not a character set. It is an encoding of the Unicode character set(*).
The confusion comes about because most other well-known encodings (eg.: ISO-8859-1) started out as separate character sets. Then when Unicode came along as a superset of most of these character sets, it became possible to think of them as different (but partial) encodings of the same (Unicode) character set, rather than just isolated character sets. Looking at them this way allows you to convert between them through Unicode easily, which would not be possible if they were merely isolated character sets. But it still makes sense to refer to them as character sets, so either term could be used.
A ‘code page’ is a term stemming from IBM, where it chose which set of symbols would be displayed. The term continued to be used by DOS and then Windows, through to Unicode-aware Windows where it just acts as an encoding with a numbered identifier. Whilst a numbered ‘code page’ is an idea not inherently limited to Microsoft, today the term would almost always just mean an encoding that Windows knows about.
When one is talking of code page ‹some number› one is typically talking about a Windows-specific encoding, as distinct from an encoding devised by a standards body. For example code page 28591 would not normally be referred to under that name, but simply ‘ISO-8859-1’. The Windows-specific Western European encoding based on ISO-8859-1 (with a few extra characters replacing some of its control codes) would normally be referred to as ‘code page 1252’.
[*: All the UTFs are encodings not character sets, but this kind of thing isn't exclusive to Unicode. For example the Japanese standard JIS X 0208 defines a character set and two different byte encodings for it: the somewhat unpleasant high-byte-based encoding (‘Shift-JIS’), and the deeply horrific escape-switching-based encoding (‘JIS’).]
Best Answer
Basically:
People sometimes use charset to refer both to the character repertoire and the encoding scheme. The Unicode Standard charset has multiple encodings, e.g., UTF-8, UTF-16, UTF-32, UCS-4, UTF-EBCDIC, Punycode, and GB18030.