Simplified Chinese vs Traditional Chinese
The term Chinese is applied to many different forms of communication. In a broad sense, the word refers to an
entire set of dialects spoken by inhabitants of China AND the written symbolic language shared by all Chinese.
In another sense, Chinese could mean specifically the written characters. Many of the dialects could well
be considered languages in their own right.
Mandarin Chinese
Many people refer to Mandarin when they say "Chinese". Since the Chinese government declared Mandarin the national language, the majority of the Chinese population has learned this dialect. Because Mandarin is the official language, it is referred to by many as Putonghua 普通話 ("the common speech"). In Taiwan, it is normally referred to as 國語. (guoyu "national language"). Mandarin is spoken by more people than every other (non-Chinese) language put together.
Written Chinese
China is divided into dozens of ethnic communities and by scores of dialects. Every province has its own local speech. Only this century has their been a common language imposed on the nation. Yet the common written language unifies the varied people and different dialects. Although they would pronounce it differently, people from different provinces understand the same character to mean the same thing and use the same character combinations to form complex ideas.
Simplified and Traditional writing
Because learning Chinese requires years of study, a large percentage of the Chinese population was illiterate. After the Chinese Cultural Revolution, many of the new government leaders decided that the current writing system was too difficult. To make learning easier, they created an alternate character set by modifying traditional characters to make them easier to write. So, there now are two forms of Chinese writing: Traditional ("Big5" character codes) and Simplified (GB2312, or "GB" character codes).
Traditional Chinese is still used to write characters in Taiwan and Hong Kong, and much of the Chinese diaspora. Simplified Chinese is used in Mainland China and Singapore. More and more publications are using simplified characters, but there are many works available only in traditional. Important note: although both Taiwan and Hong Kong use traditional Chinese, there are stylistic differences - people from Hong Kong can read Taiwanese Chinese, but may consider it incorrect. Ensure you know which target market you are aiming at!
Keep in mind that not every character has been simplified, only some of the more complicated forms. This simplification of characters did follow some logical principles. For comparison, here is a list of examples. Traditional forms are on the left, followed by their simplified forms, pinyin pronunciation, and English equivalents.
Chinese characters and Unicode
Unicode provides a superset of most character sets in use around the world, but tries not to duplicate characters unnecessarily. For example, there are several ISO character sets in the 8859 series that all duplicate the ASCII characters. Unicode doesn't have as many codes for the letter 'a' as there are character sets - that would be ridiculous. The same principle applies for Han (Chinese) characters. The initial set of sources for Han encoding in Unicode laid end to end comprised 121,000 characters, but there were many repeats, and the final Unicode tally for all these after elimination of duplicates was 20,902.
(It is said that Chinese people typically use around 3-4,000 characters for most communication, but a reasonable word processor would need to support at least 10,000. Unicode now supports over 70,000 Han characters.)
If Han characters had different meanings or etymologies, they were not unified in Unicode. Han characters, however, are highly pictorial in nature. So the (dis-)unification process had to take into account the visual forms to some extent. Where there was a significant visual difference between Han characters that represented the same thing they were allotted to separate Unicode codepoints. (This was a pretty sophisticated process, in fact, carried out over a long period by many East-Asian experts.)
The Chinese national GB standard defines a basic set of (around 6,000) characters for use with Simplified Chinese writing that does not include many of the characters in the Taiwanese industry standard for Traditional Chinese called Big 5 (around 13,000 characters in the basic set). Unicode is however a superset of both with all duplication removed down to the level of detail described above.
So the characters for 'country' in Simplified and Traditional Chinese, 国 and 國 respectively, are stored as separate codes and you cannot simply switch between the two by using a different font. On the other hand, the character for 'the world' in both Simplified and Traditional writing looks the same, 界, and both writing systems do share the same code point. Then there are characters such as 雪 ('snow') which share the code point because they are not significantly different in appearance, but may typically exhibit systematic differences in stroke overshoot and rotation of minor strokes between simplified and traditional writing systems. To see these correctly you need to apply the right font, eg. a Song font for simplified and a Ming font for traditional.
Converting between the two
There are applications that attempt to convert between simplified and traditional Chinese. (It's generally easier from traditional to simplified, since you are more likely to be mapping from many to one character than the other way around.)
Over the years, translation between simplified and traditional Chinese has been complicated by the divergences in language usage between the two communities. Grammatical differences are not generally considered to be major, but there are terminological differences such as that for the word 'computer', which is different in the PRC and Taiwan.