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.
Best Answer
UTF-8 is an encoding and that term is used in the RFC that defines it which is quoted below.
Prior to Unicode, if you wanted to use an alphabet†like Cyrillic or Greek, you needed to use a encoding that only encoded to characters in that alphabet. Thus, the terms encoding and charset were often conflated but they mean different things.
Now though, Unicode is usually the only character set you need to worry about since it contains characters for most written languages you'll have to deal with, except Klingon.
†- Alphabet, a kind of *character set* where characters correspond directly to sounds in a spoken language.A character set is a mapping from code-units (integers) to characters, symbols, glyphs, or other marks in a written language. Unicode is a character set that maps 21b integers to unicode codepoints. The Unicode Consortium's glossary describes it thus:
An encoding is a mapping from strings to strings. UTF-8 is an encoding that maps strings of bytes (8b integers) to strings of code-points (21b integers). The Unicode Consortium calls it a "character encoding scheme" and it is defined in RFC 3629.