Just as Java2 Standard Edition (J2SE) was expanded by the Java2 Enterprise Edition (J2EE), there is a smaller and more compact version of Java known as Java Micro Edition. This is a very popular platform for a variety of consumer devices such as PDAs, cell phones, etc.
In JME, "configuration" is a reserved word and at present there are two - Connected Limited Device Configuration (CLDC) and Connected Device Configuration (CDC). Configurations in this context specify the virtual machine, the base class libraries available to the API, and describe a certain class of device with regard to network connectivity, memory, etc.
"Profiles" build upon the configuration context to further define the API sufficiently for a developer to build applications.
As an example, the CLDC configuration applies to highly resource constrained devices with intermittent network connections - mobile phones, pagers, etc. You need a profile in order to target a particular class of cell phones - e.g. the Mobile Information Device Profile (MIDP) in order to target model of Nokia phones.
The configuration information, the profile, and (naturally) the custom code you write all go together determine the resource (processor, memory, etc.) demands of your application.
As an aside, these types of applications are often called MIDlets. That is a twist on Java "applet" and due to the popularity of the CLDC/MIDP combination.