Crucial in the representation of Unicode characters on your computer is to use the correct font, because the indication of a Unicode character code does not mean that the corresponding character may also be displayed on your computer. To ensure this representation, there must exist a glyph - that is, the graphical representation of a character - in the font file. Otherwise, the computer does not know how the characters should look like in the desired font. Whenever a character in a font is not present, a box is represented instead. If a character can not be stored in a encoding, a question mark is displayed instead. For example, if you save some Chinese characters as ANSI and open the file again, you can only see some question marks instead of the characters.
The more a Unicode character is exotic, the more likely it is that there is no glyph in a font for that character, even if the font is Unicode enabled. Common storage formats for fonts such as TrueType or OpenType font files can only represent 65,536 different glyphs - which are facing over 100,000 Unicode characters. So even technically it is not possible to cover all Unicode characters.
This means that the choice of font depends on the purpose. For normal purposes, such as the representation of Latin, Cyrillic, Arabic, Hebrew or Greek letters and many special characters, the standard Windows fonts such as Times New Roman, Arial and Courier New are perfectly adequate. On many systems the font Arial Unicode MS is installed, which includes glyphs for almost all major Unicode characters. In addition to the characters mentioned above, for example, glyhphs for Armenian, Devanagari, Hangul, Tibetan, Laotian, Thai, Telugu, Tamil, Oriya and Bengali, Chinese-Japanese-Korean ideographs as well as many graphical symbols are in the font file.
For more information about Unicode fonts, you can visit the following pages. The first page contains a list of many fonts and its Unicode characters, while on the other pages you can also find some other information. If you want to read about the possibilities to write Unicode characters on your computer, you should have a look at the topic Unicode Input.