Model m = ModelFactory.createDefaultModel();
String nsA = "http://somewhere/else#";
String nsB = "http://nowhere/else#";
Resource root = m.createResource( nsA + "root" );
Property P = m.createProperty( nsA + "P" );
Property Q = m.createProperty( nsB + "Q" );
Resource x = m.createResource( nsA + "x" );
Resource y = m.createResource( nsA + "y" );
Resource z = m.createResource( nsA + "z" );
m.add( root, P, x ).add( root, P, y ).add( y, Q, z );
System.out.println( "# -- no special prefixes defined" );
m.write( System.out );
System.out.println( "# -- nsA defined" );
m.setNsPrefix( "nsA", nsA );
m.write( System.out );
System.out.println( "# -- nsA and cat defined" );
m.setNsPrefix( "cat", nsB );
m.write( System.out );

The output from this fragment is three lots of RDF/XML, with three different prefix mappings. First the default, with no prefixes other than the standard ones:

# -- no special prefixes defined

<rdf:RDF
    xmlns:j.0="http://nowhere/else#"
    xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
    xmlns:j.1="http://somewhere/else#" >
  <rdf:Description rdf:about="http://somewhere/else#root">
    <j.1:P rdf:resource="http://somewhere/else#x"/>
    <j.1:P rdf:resource="http://somewhere/else#y"/>
  </rdf:Description>
  <rdf:Description rdf:about="http://somewhere/else#y">
    <j.0:Q rdf:resource="http://somewhere/else#z"/>
  </rdf:Description>
</rdf:RDF>

We see that the rdf namespace is declared automatically, since it is required for tags such as <RDF:rdf> and <rdf:resource>. Namespace declarations are also needed for using the two properties P and Q, but since their namespaces have not been introduced to the model, they get invented namespace names: j.0 and j.1.
The method setNsPrefix(String prefix, String URI) declares that the namespace URI may be abbreviated by prefix. Jena requires that prefix be a legal XML namespace name, and that URI ends with a non-name character. The RDF/XML writer will turn these prefix declarations into XML namespace declarations and use them in its output:

# -- nsA defined

<rdf:RDF
    xmlns:j.0="http://nowhere/else#"
    xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
    xmlns:nsA="http://somewhere/else#" >
  <rdf:Description rdf:about="http://somewhere/else#root">
    <nsA:P rdf:resource="http://somewhere/else#x"/>
    <nsA:P rdf:resource="http://somewhere/else#y"/>
  </rdf:Description>
  <rdf:Description rdf:about="http://somewhere/else#y">
    <j.0:Q rdf:resource="http://somewhere/else#z"/>
  </rdf:Description>
</rdf:RDF>

The other namespace still gets the constructed name, but the nsA name is now used in the property tags. There's no need for the prefix name to have anything to do with the variables in the Jena code:

# -- nsA and cat defined

<rdf:RDF
    xmlns:cat="http://nowhere/else#"
    xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
    xmlns:nsA="http://somewhere/else#" >
  <rdf:Description rdf:about="http://somewhere/else#root">
    <nsA:P rdf:resource="http://somewhere/else#x"/>
    <nsA:P rdf:resource="http://somewhere/else#y"/>
  </rdf:Description>
  <rdf:Description rdf:about="http://somewhere/else#y">
    <cat:Q rdf:resource="http://somewhere/else#z"/>
  </rdf:Description>
</rdf:RDF>

Both prefixes are used for output, and no generated prefixes are needed.
implicit prefix definitions
As well as prefix declarations provided by calls to setNsPrefix, Jena will remember the prefixes that were used in input to model.read().
Take the output produced by the previous fragment, and paste it into some file, with URL file:/tmp/fragment.rdf say. Then run the code:

Model m2 = ModelFactory.createDefaultModel();
m2.read( "file:/tmp/fragment.rdf" );
m2.write( System.out );

You'll see that the prefixes from the input are preserved in the output. All the prefixes are written, even if they're not used anywhere. You can remove a prefix with removeNsPrefix(String prefix) if you don't want it in the output.
Since NTriples doesn't have any short way of writing URIs, it takes no notice of prefixes on output and doesn't provide any on input. The notation N3, also supported by Jena, does have short prefixed names, and records them on input and uses them on output.

Jena has further operations on the prefix mappings that a model holds, such as extracting a Java Map of the exiting mappings, or adding a whole group of mappings at once; see the documentation for PrefixMapping for details.