In magento in a module if you're required to create a admin panel tab in which you need to display Category Drop down field through system.xml file in your module, then lets see how we can do it.

1- In the etc folder of the module open the system.xml(i.e., at app/code/local/Your/Module/etc ) file follow the codes as described below.

2- Now in the module in model folder create a folder structure like /System/Config/Source(i.e., at app/code/local/Your/Module/Model/System/Config/Source) in this 'Source' folder create a class category.php that extends the core class Adminhtml_Model_System_Config_Source_Category and in that class follow the codes as described at step 2.

Now, lets check out the codes for the above steps :

1- In your system.xml file add this code:

<?xml version="1.0" encoding="UTF-8"?>

<config>

    <tabs>

        <Your translate="label" module=module"">

            <label>Your Extensions</label>

            <sort_order>100</sort_order>

        </Your>

    </tabs>

    <sections>

        <Your translate="label" module="module">

            <label> Module Options</label>

            <tab>Your</tab>

            <sort_order>1000</sort_order>

            <show_in_default>1</show_in_default>

            <show_in_website>1</show_in_website>

            <show_in_store>1</show_in_store>


            <groups>

                <your_group translate="label" module="module">

                    <label>General</label>

                    <frontend_type>text</frontend_type>

                    <sort_order>1000</sort_order>

                    <show_in_default>1</show_in_default>

                    <show_in_website>1</show_in_website>

                    <show_in_store>1</show_in_store>


                    <fields>


                        <your_field_name translate="label comment">

                            <label>Category: </label>

                            <comment>Choose category.</comment>

                            <frontend_type>select</frontend_type>

                            <source_model>Your_module/system_config_source_category</source_model>

                            <sort_order>2</sort_order>

                            <show_in_default>1</show_in_default>

                            <show_in_website>0</show_in_website>

                            <show_in_store>0</show_in_store>

                        </your_field_name>


                    </fields>

                </your_group>

            </groups>

        </Your>

    </sections>

</config>

Suppose, in the section as 'your' you have a group as 'your_group' in which you need to add a field 'Category'. For that purpose in the fields tag create a field as 

<module_choose_category translate="label comment">

<label>Category: </label>

<comment>Choose category.</comment>

<frontend_type>select</frontend_type>

<source_model>module/system_config_source_category</source_model>

<sort_order>2</sort_order>

<show_in_default>1</show_in_default>

<show_in_website>0</show_in_website>

<show_in_store>0</show_in_store>

</module_choose_category>  

In that field we have used two main important tags that are <frontend_type> and <source_model>.

The tag <frontend_type> defines the input type of the field as i have set it to "select" which is of drop down.

And the <source_model> tag contains the path of the model class with the path defined as module/system_config_source_category.

2- Now in the module in model in the class Your_Module_Model_System_Config_Source_Category add this function toOptionArray.

<?php

class Your_Module_Model_System_Config_Source_Category extends Mage_Adminhtml_Model_System_Config_Source_Category

{

    public function toOptionArray($addEmpty = true)

    {

        $collection = Mage::getModel('catalog/category')->getCollection()->addAttributeToSelect('*');


        $options = array();      

        foreach($collection as $category){

            if($category->getName() != 'Root Catalog'){

                $options[] = array(

                   'label' => $category->getName(),

                   'value' => $category->getId()

                );

            }

        }

        return $options;

    }

}

In above code the class extends Mage_Adminhtml_Model_System_Config_Source_Category and override the function toOptionArray as above to get the categories in the $options array and return it.

In this way the category dropdown field will be created on the admin panel through your system.xml file in the your desired section.