> I imagine that I need
> to call another template such as "third" below, 

You're making it look much more complicated than it is:


<xsl:stylesheet xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
                version="1.0"
                >

<xsl:output method="html" indent="yes"/>

<xsl:template match="DATA">
 <table>
  <xsl:apply-templates select="HISTORY"/>
 </table>
</xsl:template>


<xsl:template match="HISTORY">
  <tr>
  <td><xsl:apply-templates select="DESCRIPTION"/></td>
  <td><xsl:apply-templates select="../NAME[@CODE=current()/NAME]/PERSON"/></td>
  </tr>
</xsl:template>


</xsl:stylesheet>

produces
<table>
   <tr>
      <td>This is a history item 1</td>
      <td>Fred Bloggs</td>
   </tr>
   <tr>
      <td>This is a history item 2</td>
      <td>John Smith</td>
   </tr>
   <tr>
      <td>This is a history item 3</td>
      <td>Fred Bloggs</td>
   </tr>
   <tr>
      <td>This is a history item 4</td>
      <td>Fred Jones</td>
   </tr>
   <tr>
      <td>This is a history item 5</td>
      <td>Fred Bloggs</td>
   </tr>
</table>

on your input example. (which is what you asked for except that I
downcased the html elements: that future proofs the code for moving to
xhtml which is all lowercase.)

 
David


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