I think I see the problem. When you populate $name, the variable name is lowercase. However, when you do anything with it, it's $NAME, or uppercase. Try doing everything in lowercase and see if it works.

In addition, as scain said, make sure you're typing everything in the same case. See the version of the script I put below for a way to make the script case-insensitive.

A few notes:

1. If you're typing within the <code> tags, you don't need to use any other HTML tags, specifically <br>.
2. chomp is better than chop, for the purposes you're putting it to. (Getting rid of the newline character(s). If you're on a Windows machine, that may be the reason you're having problems. That script was written for a Unix machine, originally.)
3. Although Learning Perl doesn't get into this until a little later, you should start getting into the habit of using strict and warnings. There are a number of nodes on PerlMonks that address this. If I was writing your script from scratch, I'd write it as such:

   #!/usr/bin/perl -w
   print "What is your name? ";
   my $name = <STDIN>;
   chomp($name);
   if (uc($name) eq 'RANDAL') {
       print "Hello, Randal! How good of you to be here!\n";
   } else {
       print "Hello, $name\n";
   }
   
   __END__
   [download]

Update: Added a few comments after reading scain's response. Added uc to the script listing.

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/me wants to be the brightest bulb in the chandelier!