The data you have is JSON. And a module such as JSON has decode_json to convert it to a Perl data structure. Alternatively, if you don't want to install JSON, Perl ships with JSON::PP, which is a drop-in replacement (or fall-back).

#!/usr/bin/env perl

use strict;
use warnings;
use JSON qw(decode_json);

my $json = do {
    local $/ = undef;
    <DATA>;
};

my $data = decode_json($json);

foreach my $element (@{$data->{'genres'}}) {
    print "id: $element->{'id'}, name: $element->{'name'}\n";
}

__DATA__
{
  "genres": [
    {
      "id": 28,
      "name": "Action"
    },
    {
      "id": 12,
      "name": "Adventure"
    },
    {
      "id": 16,
      "name": "Animation"
    },
    {
      "id": 35,
      "name": "Comedy"
    },
    {
      "id": 80,
      "name": "Crime"
    },
    {
      "id": 99,
      "name": "Documentary"
    },
    {
      "id": 18,
      "name": "Drama"
    },
    {
      "id": 10751,
      "name": "Family"
    },
    {
      "id": 14,
      "name": "Fantasy"
    },
    {
      "id": 36,
      "name": "History"
    },
    {
      "id": 27,
      "name": "Horror"
    },
    {
      "id": 10402,
      "name": "Music"
    },
    {
      "id": 9648,
      "name": "Mystery"
    },
    {
      "id": 10749,
      "name": "Romance"
    },
    {
      "id": 878,
      "name": "Science Fiction"
    },
    {
      "id": 10770,
      "name": "TV Movie"
    },
    {
      "id": 53,
      "name": "Thriller"
    },
    {
      "id": 10752,
      "name": "War"
    },
    {
      "id": 37,
      "name": "Western"
    }
  ]
}
[download]

This produces:

id: 28, name: Action
id: 12, name: Adventure
id: 16, name: Animation
id: 35, name: Comedy
id: 80, name: Crime
id: 99, name: Documentary
id: 18, name: Drama
id: 10751, name: Family
id: 14, name: Fantasy
id: 36, name: History
id: 27, name: Horror
id: 10402, name: Music
id: 9648, name: Mystery
id: 10749, name: Romance
id: 878, name: Science Fiction
id: 10770, name: TV Movie
id: 53, name: Thriller
id: 10752, name: War
id: 37, name: Western
[download]

A foreach loop is useful here, as described in perlintro or perlsyn. Dealing with complex data structures is covered in Intermediate Perl (OReilly), in perlreftut, perlref, perldsc, and perllol.