Leopard (OS X 10.5) was the first version of OS X to make the Java 6 (1.6) JRE available, through an update. Unfortunately, only the 64-bit version of the Java 6 JRE is available for Leopard, so only those with 64 bit Intel processors can run Java 1.6.
Since the Core 2 Duo, all available Intel Macs are 64-bit capable. If you have a Core Duo-based Intel mac (note the missing 2), your computer is not 64-bit capable. The first Macbook (Pro) and first Mac mini are examples of that. Your particular computer is 32-bit, so you don't won't get Java 6 with Leopard, and certainly don't have it with Tiger (10.4).
Snow Leopard (OS X 10.6) comes with Java 6 out of the box. So, if you get a new Mac with Snow Leopard, or you upgrade to Snow Leopard, it includes both a 32-bit and 64-bit version of Java 6. If you upgrade your 32-bit Intel Mac to Snow Leopard, you will get Java 6 (32-bit).
For those Intel Mac owners who do not wish to upgrade to Snow Leopard, or cannot upgrade (Snow Leopard requires 1 GB of RAM to install), you have the option of installing SoyLatte for Java 6.
SoyLatte (http://landonf.bikemonkey.org/static/soylatte/) is derived from the FreeBSD port of the Java 6 JRE and runs on the Mac. SoyLatte is only for Intel based Macs, and only includes the 32-bit JRE.
Simply download and extract SoyLatte, set your JAVA_HOME and PATH, and run your program. Here's a sample:
$ export JAVA_HOME=/Users/madams/Desktop/soylatte16-i386-1.0.3
$ export PATH=/Users/madams/Desktop/soylatte16-i386-1.0.3/bin:$PATH
$ java -jar /Users/madams/Desktop/test.jar
Also, note that Snow Leopard includes only Java 6, and automatically removes older JREs. This means, no more Java 1.5, though the paths for previous versions are linked to the current one.
In addition, Java 6 in Snow Leopard defaults to the 64-bit JRE on 64-bit Macs, so if you are calling 32-bit native libraries (JNIs), it will fail. You must call java using the -d32 command line option to run the 32-bit version of the Java JRE on a 64-bit Mac with Snow Leopard.