gistfile1.php

<?php $cat_id = get_query_var('cat');
				
				if( ! empty( $cat_id ) ) {
					$category = get_category( $cat_id, ARRAY_A );
				
					if( ! empty( $category ) ) {
						// parent category
						if( $category['parent'] === '0' ) {
							
							// get child IDs
							$terms = get_term_children( $cat_id, 'category' );
							
							foreach( $terms as $term_id ) {
								$child_category = get_term( $term_id, 'category' );
								
								// if you need the category
								$category_name = $child_category->name;
								
								
								/**
								 * Sample data in your child category object: 
								 * 
								 * object(stdClass)[365]
										  public 'term_id' => string '14' (length=2)
										  public 'name' => string 'Child 1' (length=7)
										  public 'slug' => string 'child-1' (length=7)
										  public 'term_group' => string '0' (length=1)
										  public 'term_taxonomy_id' => string '14' (length=2)
										  public 'taxonomy' => string 'category' (length=8)
										  public 'description' => string '' (length=0)
										  public 'parent' => string '12' (length=2)
										  public 'count' => string '5' (length=1)
										  public 'object_id' => string '33' (length=2)
								 * 
								 * 
								 */
								
								// do whatever you like with the categories now...
							}
							
						} else { // in child category
							// do the regular loop here, if ( have_posts() ) ...
						}
					}
				}