The easiest way to upgrade to the latest version of ImageJ is to use the Plugins>Utilities>Update ImageJ command, available in version 1.39 or later. Use the ImageJ Updater plugin to upgrade earlier versions. You can also upgrade by downloading the latest JAR file (ij.jar) from rsb.info.nih.gov/ij/upgrade/ij.jar and copying it into the ImageJ folder. The Java source code is available at rsb.info.nih.gov/ij/download/src/.
Note 1: ImageJ 1.37 requires Java 1.3.1 or later. ImageJ 1.38 requires Java 1.4.1 or later.
Note 2: Internet Explorer has the nasty habit of sometimes renaming ij.jar. It may tell you it's saving 'ij.jar', but what you get is 'ij.jar.zip'. Just find that file and rename it 'ij.jar', and you're all set.
Note 3: To succesfully download the ij.jar file, you may need to right-click on the link and select "Save Link As...".