Pre OS X

Mac users have it easy, so I won't mention much here.  If you have a version of Mac OS before OS X, then there is a special Japanese version of Mac OS called KanjiTalk you need to get and install for full Japanese support.  If you just want to display Japanese web pages, then this isn't necessary, and you can use any normal web browser, so long as the "text encoding" extension is installed.  If the text doesn't show up right, try fiddling with the "encoding" menu.  Likewise, if the text show sup as boxes, make sure you have a Japanese font installed, and the web browser is configured to use it.

OS X

All versions of OS X to date are effectively international.  You can view Japanese web pages as explained above.  To enable Japanese input support, go to the "Input Sources" tab of the languages preference panel and make sure Japanese input is enabled.  (Check Katakana, Hiragana, etc.)

Interface Display Language

If you have Leopard (10.5), simply go to the languages pane of system preferences application, and drag Japanese to the top, and from then on application menus will be displayed in Japanese.  If Japanese isn't in the list of options, just click the "Edit List..." button to add it.  (Check out a screen-cast of the process here).  With Snow Leopard (10.6), Apple stopped installing all languages on every system by default to save disk space.  If Japanese support isn't installed, simply install it and then follow the instructions above.  Note that you can use Japanese input while the system language is set to English or another language if you would like. 

This varies by distribution and version, but so long as Japanese fonts are installed, any version should be able to view Japanese web pages fine.  Likewise, viewing application menus in Japanese simply requires selecting the Japanese locale when logging in.  If This doesn't work, there are probably some Japanese packages you need to install.  Japanese input can be more complicated, as there is typically some daemon that needs to be running, and it may automatically start only when you log in using Japanese.  Consult your distribution's documentation.

Pre-2000

Versions of Windows before Windows 2000 were available only in localized versions, and the user interface simply couldn't be switched, also the multi-language handling was very weak, as the entire OS handled text internally according to the language it was built for.  If you want to do Japanese in Windows 98, for example, you basically need the Japanese edition of Windows 98.  You may be able to install Japanese fonts and get web pages to work, though.  Likewise Japanese input support is available, but may only work in certain applications (like Microsoft office and outlook), as opposed to system-wide.

2000, XP, Vista, and Windows 7

Windows 2000 and onward are global in that they handle text internally with Unicode, meaning they can handle any and all languages simultaneously - at least theoretically.  If you want to install Japanese-only applications in an English version of Windows, you are better off switching the non-Unicode locale to Japanese in the control panel.  If you don't do this, then legacy applications using SJIS instead of Unicode will display garbage on the screen.

You can install Japanese support (fonts, etc.) in the control panel as well (this will require your Windows CD), and then you will be able to display Japanese web pages and other documents.  If web pages don't display properly, then check the encoding options, and make sure it is set to "Japanese (auto)" in your web browser of choice.  Sometimes you will have to force it to a particular encoding if the web browser's automatic guess is wrong.  

After you install Japanese locale support, you can additionally install Input support (also from the control panel), which will give you an input bar.  If you don't have a Japanese keyboard with the special mode-switching keys, you will need to check the key combination for switching between Japanese and English entry modes.  Usually it is something like Alt-~ or Alt-Shift.

The final level is the most difficult.  Most Windows applications are built with the interface in only one language.  If you install the English version of Firefox, the menus will be in English.  If you want a Japanese user interface, then you have to install the Japanese version of Firefox (and there is no way to make it show in English).  This is typical of most Windows programs, but there are exceptions.  Skype, for example, contains multiple languages, and there is a menu to switch between them at any time.

As for the OS itself (including Internet Explorer), Windows can theoretically support multiple interface languages, but it isn't a common consumer option.  If you have Windows XP professional and can get ahold of the "MUI" installers, you can install Japanese, set your user interface to Japanese, and then log back in, and you will have what looks like Japanese Windows.  Other users can use English, and you can change your setting at any time (provided you don't mind logging out and back in each time you change the language).  

With Windows 7, the MUI pack is easier to get, as it can be downloaded from Windows update - but only if you shelled out extra for the Ultimate edition.  Figuring out exactly why Windows is behind the curve on this is an exercise I will leave to the readers.  Note that Windows 7 Professional can be coerced into using MUI if you can find the Japanese MUI package, and then install and configure it manually instead of through the control panel.  (Instructions for this can be found on various web sites).  With Windows XP, only the English version can use MUI.  (i.e. you can add a Japanese interface to the English version, but not vice versa).  It appears that with Windows 7, all versions of windows are really the same, they just normally include only one language pack.  

All of the operating systems have a standard IME system for most languages, but there are third party systems available that are smarter and more accurate.  The most popular commercial third party IME is called ATOK.  Aside from ATOK, Google has now released their own Japanese IME for Windows and Mac OS (for free).  To find it, simply search for グーグル日本語入力 in Google.  The advantage of using these systems is that they require less tweaking to get the right sentence oftentimes, and can predict the sentences better than the standard IME systems, much like the Japanese input on mobile phones.