How to remove elements from xml using xslt with stylesheet and xsltproc?

Using one of the most fundamental XSLT design patterns: "Overriding the identity transformation" one will just write the following:

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

 <xsl:output omit-xml-declaration="yes"/>

    <xsl:template match="node()|@*">
      <xsl:copy>
         <xsl:apply-templates select="node()|@*"/>
      </xsl:copy>
    </xsl:template>

    <xsl:template match="Element[@fruit='apple' and @animal='cat']"/>
</xsl:stylesheet>

Do note how the second template overrides the identity (1st) template only for elements named "Element" that have an attribute "fruit" with value "apple" and attribute "animal" with value "cat". This template has empty body, which means that the matched element is simply ignored (nothing is produced when it is matched).

When this transformation is applied on the following source XML document:

<doc>... 
    <Element name="same">foo</Element>...
    <Element fruit="apple" animal="cat" />
    <Element fruit="pear" animal="cat" />
    <Element name="same">baz</Element>...
    <Element name="same">foobar</Element>...
</doc>

the wanted result is produced:

<doc>... 
    <Element name="same">foo</Element>...
    <Element fruit="pear" animal="cat"/>
    <Element name="same">baz</Element>...
    <Element name="same">foobar</Element>...
</doc>

More code snippets of using and overriding the identity template can be found here.

The answer by @Dimitre Novatchev is certainly both correct and elegant, but there's a generalization (that the OP didn't ask about): what if the element you want to filter also has child elements or text that you want to keep?

I believe this minor variation covers that case:

<?xml version="1.0" encoding="UTF-8"?>
<xsl:stylesheet xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
    xmlns:xs="http://www.w3.org/2001/XMLSchema"
    version="2.0">

    <xsl:template match="node()|@*">
        <xsl:copy>
            <xsl:apply-templates select="node()|@*"/>
        </xsl:copy>
    </xsl:template>

    <!-- drop DropMe elements, keeping child text and elements -->
    <xsl:template match="DropMe">
        <xsl:apply-templates/>
    </xsl:template>

</xsl:stylesheet>

The match condition can be complicated to specify other attributes, etc., and you can use multiple such templates if you're dropping other things.

So this input:

<?xml version="1.0" encoding="UTF-8"?>
<mydocument>
    <p>Here's text to keep</p>
    <p><DropMe>Keep this text but not the element</DropMe>; and keep what follows.</p>
    <p><DropMe>Also keep this text and <b>this child element</b> too</DropMe>, along with what follows.</p>
</mydocument>

produces this output:

<?xml version="1.0" encoding="UTF-8"?><mydocument>
    <p>Here's text to keep</p>
    <p>Keep this text but not the element; and keep what follows.</p>
    <p>Also keep this text and <b>this child element</b> too, along with what follows.</p>
</mydocument>

Credit to XSLT Cookbook.