This week's book giveaway is in the OO, Patterns, UML and Refactoring forum. We're giving away four copies of Refactoring for Software Design Smells: Managing Technical Debt and have Girish Suryanarayana, Ganesh Samarthyam & Tushar Sharma on-line! See this thread for details.
On normal (Intel-based) PCs, you can run 64-bit as well as 32-bit software if you have a 64-bit operating system.
Note that with Eclipse, the version of Eclipse that you use has to match your Java version - if you have 32-bit Java, you must use 32-bit Eclipse, and if you use 64-bit Java, you must use 64-bit Eclipse. That is because Eclipse uses some native libraries (for SWT), and Java can only load those native libraries if they have the same "bitness" as Java itself.
For Java bytecode, it doesn't matter what the "bitness" of the JDK is. The class files produced by a 32-bit JDK are exactly the same as those produced by a 64-bit JDK. So you can compile on 32-bit and run on 64-bit or vice versa. Java bytecode is independent of the "bitness" of the JDK and the OS.