First of all, we have to create a module for adding custom category attribute.

Step 1. Create a new module.
We should let Magento know about our new module. The initial configuration file is located at ‘app/etc/modules/Namespace_Modulename.xml’.
Namespace_Modulename.xml

<?xml version="1.0"?>
<config>
    <modules>
        <Namespace_Modulename>
            <active>true</active>
            <codePool>local</codePool>
        </Namespace_Modulename>
    </modules>
</config>

Step 2.
Module configuration file is located at ‘app/code/<code_pool>/<name_space>/<module_name>/etc’ and its name is config.xml – so this path looks like ‘app/code/local/Namespace/Modulename/etc/config.xml’.

<?xml version="1.0"?>
<config>
    <modules>
        <Namespace_Modulename>
            <version>1.0.0</version>
        </Namespace_Modulename>
    </modules>

    <global>
        <resources>
            <add_category_attribute>
                <setup>
                    <module>Namespace_Modulename</module>
                    <class>Mage_Catalog_Model_Resource_Setup</class>
                </setup>
                <connection>
                    <use>core_setup</use>
                </connection>
            </add_category_attribute>
            <add_category_attribute_write>
                <connection>
                    <use>core_write</use>
                </connection>
            </add_category_attribute_write>
            <add_category_attribute_read>
                <connection>
                    <use>core_read</use>
                </connection>
            </add_category_attribute_read>
        </resources>
    </global>
</config>

<add_category_attribute> presence that script must be located in the folder with the same name. ‘app/code/local/Namespace/Modulename/sql/add_category_attribute’

Step 3.
Create and install a script file in the folder ‘add_category_attribute’, and the file name depends on the module version.
mysql4-install-1.0.0.php

<?php
$this->startSetup();
$this->addAttribute(Mage_Catalog_Model_Category::ENTITY, 'attribute_custom', array(
    'group'         => 'General Information',
    'input'         => 'textarea',
    'type'          => 'text',
    'label'         => 'Custom attribute',
    'backend'       => '',
    'visible'       => true,
    'required'      => false,
    'visible_on_front' => true,
    'global'        => Mage_Catalog_Model_Resource_Eav_Attribute::SCOPE_STORE,
));

$this->endSetup();
?>