There are multiple sources for the Oracle JRE:
Java for Windows is available in both 32 bit and 64 bit editions. Go with the 32 bit version by default, or see Which Java download should I choose for my 64-bit Windows operating system?
During the installation, it may say that "Java content is disabled in web browsers". From a security point of view this is a good thing, as all the security issues with Java come from applets in web pages. If the use of Java in browsers was disabled when Java was un-installed, a record of this is kept nonetheless (only tested on Windows). A new install of Java will see that you previously had it disabled in browsers and ask to confirm that's still what you want.
On Windows 7, Java runs a program at system startup to check for updates to itself. The program is called SunJavaUpdateSched and it is jusched.exe.
If you need to update Java, I suggest un-installing the old version manually, then downloading the offline version. This way, you get to confirm that the un-install of the old version worked before starting the installation of the new version, and, it makes the new version installation simpler.
The above recommendation goes way back, but on Jan. 22, 2013 Ed Bott wrote about another reason not to let Java self-update: it tries to install extra software that hijacks your web searches.
On Windows machines, Java is un-installed in the normal manner from the Control Panel. On XP with "Add/Remove Programs," on Windows 7 it's "Programs and Features". Now, Java is actually called Java, but in the dim old days it was listed in the Control Panel as "J2SE Runtime Environment". The bad old days also used to accumulate old versions of Java. Now, they should be removed when a new version is installed. Still, on an older PC, it's a good idea to verify that all versions of Java have been removed before installing a new version.
On Windows XP, you can safely disable the Java quick starter service. This needs to be done after each new version installation.
Finally, there is the issue of updating Java. One reason bad guys attack Java is that updates roll out to end users slooooooooowly. In part, this is because Java's self-update checking is/was not done frequently enough. Defaults change over time, but for a long time, Java defaulted to checking for updates once a month. The current default is confusing. As best as I can understand it, Java now (as of Dec. 2012) checks for updates once a week, but only tells you about new versions once a month.
If you want Java to frequently check for updates, then after a new installation of Java, go into the Java thingy in the Windows Control Panel and configure a schedule on the Update tab. The most frequent option is daily. You will need to be logged on as an administrator to change the update schedule.
Not everyone wants, Java to check for updates. This too, can be configured on the Update tab, with one exception. Turning off all update checking does not prevent program jusched.exe from running at boot time. That needs to be done separately.
Java 7 from Oracle for OS X 10.7 and 10.8 can be downloaded at java.com/en/download/manual.jsp. There is no Java 7 from Apple.
Java 6 for OS X 10.6 is available from Apple using the normal OS update mechanism.