@ARGV = qw(file2);
my %names_in_file2;
while (<>) {
  my ($name) = split /\|/;
  $names_in_file2{$name}++;
}
@ARGV = qw(file1);
$^I = ".bak"; # create backup file
while (<>) { #  in-place edit
  my ($name) = split /\|/;
  print unless $names_in_file2{$name};
}
[download]

Now that's the way to answer questions you suspect as homework. Using a solution that nobody would generally use under normal circumstances. :)

You don't even need perl for this, the unix text utilities are powerful enough.

If you don't mind that your files get sorted, you can use

join -t\| -v 1 <(sort first) <(sort second)
[download]

(I don't want to cheat by using awk)

Update 2009 sep 2.

See Re^2: Joining two files on common field for a list of other nodes where unix textutils is suggested to merge files.