If you want to support a variety of languages for web content, you should use an encoding that covers the entire Unicode range. The best choice for this purpose is UTF-8. UTF-8 is the preferred encoding for the web; from the HTML5 draft standard:
Authors are encouraged to use UTF-8. Conformance checkers may advise authors against using legacy encodings. [RFC3629]
Authoring tools should default to using UTF-8 for newly-created documents. [RFC3629]
UTF-8 and Windows-1252 are the only encodings required to be supported by browsers, and UTF-8 and UTF-16 are the only encodings required to be supported by XML parsers. UTF-8 is thus the only common encoding that everything is required to support.
The following is more of an expanded response to Liv's answer than an answer on its own; it's a description of why UTF-8 is preferable to UTF-16 even for CJK content.
For characters in the ASCII range, UTF-8 is more compact (1 byte vs 2) than UTF-16. For characters between the ASCII range and U+07FF (which includes Latin Extended, Cyrillic, Greek, Arabic, and Hebrew), UTF-8 also uses two bytes per character, so it's a wash. For characters outside the Basic Multilingual Plane, both UTF-8 and UTF-16 use 4 bytes per character, so it's a wash there.
The only range in which UTF-16 is more efficient than UTF-8 is for characters from U+07FF to U+FFFF, which includes Indic alphabets and CJK. Even for a lot of text in that range, UTF-8 winds up being comparable, because the markup of that text (HTML, XML, RTF, or what have you) is all in the ASCII range, for which UTF-8 is half the size of UTF-16.
For example, if I pick a random web page in Japanese, the home page of nhk.or.jp, it is encoded in UTF-8. If I transcode it to UTF-16, it grows to almost twice its original size:
$ curl -o nhk.html 'http://www.nhk.or.jp/'
$ iconv -f UTF-8 -t UTF-16 nhk.html > nhk.16.html
$ ls -al nhk*
-rw-r--r-- 1 lambda lambda 32416 Mar 13 13:06 nhk.16.html
-rw-r--r-- 1 lambda lambda 18337 Mar 13 13:04 nhk.html
UTF-8 is better in almost every way than UTF-16. Both of them are variable width encodings, and so have the complexity that entails. In UTF-16, however, 4 byte characters are fairly uncommon, so it's a lot easier to make fixed width assumptions and have everything work until you run into a corner case that you didn't catch. An example of this confusion can be seen in the encoding CESU-8, which is what you get if you convert UTF-16 text into UTF-8 by just encoding each half of a surrogate pair as a separate character (using 6 bytes per character; three bytes to encode each half of the surrogate pair in UTF-8), instead of decoding the pair to its codepoint and encoding that into UTF-8. This confusion is common enough that the mistaken encoding has actually been standardized so that at least broken programs can be made to interoperate.
UTF-8 is much smaller than UTF-16 for the vast majority of content, and if you're concerned about size, compressing your text will always do better than just picking a different encoding. UTF-8 is compatible with APIs and data structures that use a null-terminated sequence of bytes to represent strings, so as long as your APIs and data structures either don't care about encoding or can already handle different encodings in their strings (such as most C and POSIX string handling APIs), UTF-8 can work just fine without having to have a whole new set of APIs and data structures for wide characters. UTF-16 doesn't specify endianness, so it makes you deal with endianness issues; actually there are three different related encodings, UTF-16, UTF-16BE, and UTF-16LE. UTF-16 could be either big endian or little endian, and so requires a BOM to specify. UTF-16BE and LE are big and little endian versions, with no BOM, so you need to use an out-of-band method (such as a Content-Type HTTP header) to signal which one you're using, but out-of-band headers are notorious for being wrong or missing.
UTF-16 is basically an accident, that happened because people thought that 16 bits would be enough to encode all of Unicode at first, and so started changing their representation and APIs to use wide (16 bit) characters. When they realized they would need more characters, they came up with a scheme for using some reserved characters for encoding 32 bit values using two code units, so they could still use the same data structures for the new encoding. This brought all of the disadvantages of a variable-width encoding like UTF-8, without most of the advantages.