Before we see what Unicode is, it makes sense to step back slightly to think about just what it means to store "characters" in digital files. Anyone who uses a tool like a text editor usually just thinks of what they are doing as entering some characters numbers, letters, punctuation, and so on. But behind the scene a little bit more is going on. "Characters" that are stored on digital media must be stored as sequences of ones and zeros, and some encoding and decoding must happen to make these ones and zeros into characters we see on a screen or type in with a keyboard.
Sometime around the 1960s, a few decisions were made about just what ones and zeros (bits) would represent characters. One important choice that most modern computer users give no thought to was the decision to use 8-bit bytes on nearly all computer platforms. In other words, bytes have 256 possible values. Within these 8-bit bytes, a consensus was reached to represent one character in each byte. So at that point, computers needed a particular encoding of characters into byte values; there were 256 "slots" available, but just which character would go in each slot? The most popular encoding developed was Bob Bemers' American Standard Code for Information Interchange (ASCII), which is now specified in exciting standards like ISO-14962-1997 and ANSI-X3.4-1986(R1997). But other options, like IBM's mainframe EBCDIC, linger on, even now.
ASCII itself is of somewhat limited extent. Only the values of the lower-order 7-bits of each byte might contain ASCII-encoded characters. The top 7-bits worth of positions (128 of them) are "reserved" for other uses (back to this). So, for example, a byte that contains "01000001" might be an ASCII encoding of the letter "A", but a byte containing "11000001" cannot be an ASCII encoding of anything. Of course, a given byte may or may not actually represent a character; if it is part of a text file, it probably does, but if it is part of object code, a compressed archive, or other binary data, ASCII decoding is misleading. It depends on context.
The reserved top 7-bits in common 8-bit bytes have been used for a number of things in a character-encoding context. On traditional textual terminals (and printers, etc.) it has been common to allow switching between codepages on terminals to allow display of a variety of national-language characters (and special characters like box-drawing borders), depending on the needs of a user. In the world of Internet communications, something very similar to the codepage system exists with the various ISO-8859-* encodings. What all these systems do is assign a set of characters to the 128 slots that ASCII reserves for other uses. These might be accented Roman characters (used in many Western European languages) or they might be non-Roman character sets like Greek, Cyrillic, Hebrew, or Arabic (or in the future, Thai and Hindi). By using the right codepage, 8-bit bytes can be made quite suitable for encoding reasonable sized (phonetic) alphabets.
Codepages and ISO-8859-* encodings, however, have some definite limitations. For one thing, a terminal can only display one codepage at a given time, and a document with an ISO-8859-* encoding can only contain one character set. Documents that need to contain text in multiple languages are not possible to represent by these encodings. A second issue is equally important: Many ideographic and pictographic character sets have far more than 128 or 256 characters in them (the former is all we would have in the codepage system, the latter if we used the whole byte and discarded the ASCII part). It is simply not possible to encode languages like Chinese, Japanese, and Korean in 8-bit bytes. Systems like ISO-2022-JP-1 and codepage 943 allow larger character sets to be represented using two or more bytes for each character. But even when using these language-specific multibyte encodings, the problem of mixing languages is still present.