The most common problem with installing Java and Eclipse occurred on machines running Vista or Windows 7 (64-bit versions) are: 1) The JRE has dependencies in the JDK, so you should install the entire JDK and not just the JRE; and 2) Eclipse can't find Java.

To fix problem 1:
Go to http://www.oracle.com/technetwork/java/javase/downloads/index.html
Click on the "Java" download button on the left
Click "Download"
Select your computer's platform from the drop-down menu, scroll down, and click "Continue"
Click the underlined .exe line under "File Description and Name"
Save this file to your Desktop
When the file has finished downloading, double-click on it and continue through the installation instructions. This should install Java on your computer and put relevant Java files at C:\Program Files\Java

To fix problem 2:
There are two ways to do this. One is easier, but more of a hack. The other way is what the computer would like you to do, but it's a little trickier.
Easier way:
Go to the place where the Java files have been installed (Something like C:\Program Files\Java)
Right-click on the folder titled "jre" or "jre6" and select Copy
Open the folder where you downloaded Eclipse
Right-click and select Paste
If the pasted folder is named "jre6", rename it to "jre"

Harder way:
Click on the start button in the lower left hand order of the screen (It should say "Start" or show a Windows icon)
Right-click on "Computer" and select "Properties"
Click on "Advanced System Settings"
Click on "Environment Variables"
In the bottom window, scroll down until you see "Path", select it, and click "Edit"
Go to the very end of the line of text, type a semicolon, then paste in the path to the jre bin folder, something like C:\Program Files\Java\jre6\bin
Click "OK", then "OK"