Get all active category names.

Magento_Example.php

<?php

public function getAllCategoryNames()
{
    $names = array();

    // Build out a category collection and add the name attribute to the select.
    $model = Mage::getModel('catalog/category');
    $collection = $model->getCollection();
    $collection->addAttributeToSelect('name');

    // Make sure the categories are actually active.
    $collection->addIsActiveFilter();

    // Start at the root.
    $collection->addLevelFilter(1);

    // Order by name (obviously, optional).
    $collection->addOrderField('name');

    foreach ($collection as $category) {
        /* @var $category Mage_Catalog_Model_Category */
        $names[] = $category->getName();
    }

    return $names;
}

public function getAllCategoryNamesRespectDepth()
{
    $names = array();
    $helper = Mage::helper('catalog/category');
    // Watch out, this method respects the depth limit set in the
    // 'catalog/navigation/max_depth' setting in admin.
    $collection = $helper->getStoreCategories('name', true, false);

    foreach ($collection as $category) {
        /* @var $category Mage_Catalog_Model_Category */
        $names[] = $category->getName();
    }

    return $names;
}