First of all, you should use strict; and use warnings; if you want to learn Perl the easy way

There are some errors in your code, some of them:

• file1.txt and file2.txt are literal strings, so you should put them in quotation marks
• After read a file, you should chomp the line/s read
• The syntax you are using for grep is not correct, you should be using grep BLOCK LIST

Here is a working version of your program:

use strict;
use warnings;

open(FH1,"file1.txt");
open(FH2,"file2.txt");
my @array=<FH2>;
chomp @array;

while(my $var=<FH1>){
  chomp $var;
  my ($first,$second)=split(/\s+/,$var);
  my @grepped= grep {/\Q$second\E/} @array;
  print "$first\t$grepped[0]\n";
}
[download]

You should also consider using hashes to solve your problem

Another alternative, if the script is only going to join both files and output the result (and you are in a Unix/Linux system, as it seems to be the case) is to use bash's join tool:

$ join -1 2 -2 1 file1.txt file2.txt
[download]

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