There are two main types of Java platforms that can be made available on OpenMoko devices such as the Neo1973.
This is the incarnation of the desktop version of Sun's Java platform. The majority of its codebase was recently opensourced under the GPL+exception license; the closed bits are quickly being replaced by unencumbered alternatives.
When people talk of Java applications, they're usually referring to software targetted to this platform.
This is the 'lite' edition of the Java platform. Modular and optimised for embedded devices, this platform provides a much more restricted set of language and library features.
When talking about games, and mobile phone games in general, people are normally referring to this platform.