It sounds like you have a CSV file, for which you may want to use a CSV module. One straight-forward-to-use and well-vetted one is Text::CSV. While this may be a simple case, frequently it is the simple stuff that you have to keep returning to and reusing...

I do not see the matching you claim in the post. Is this simple equality, or is there a more subtle pattern? Assuming simple equality, this is a good opportunity to use hashes to organize your data. If you are unfamiliar with hashes, see Perl variable types. If you are having problems reading in the file in the first place, check out Files and I/O. One implementation that does what I think you want is:

#!/usr/bin/perl
use strict;
use warnings;

my %data = ();
while (<DATA>) {
    my ($term1, $term2) = split;
    if (exists $data{$term2}) {
        print "$term2 found in both columns\n";
    }
    $data{$term1} = $term2;
}

__DATA__
NP_041954.1 NP_848263.1
NP_041955.1 NP_041956.1
I_match Not_here_though
I_dont_match I_match
[download]

Note I test before I store the value so that the entries must be in different rows (the spec as I understand it).