Who is Durgā?
The word Durga stands for a demon, who was a great obstruction representing human bondage, the deeds of the universe, grief, pain, hell, Yamaḍaṇḍa, birth, great danger and incurable disease. Therefore the one who destroys all these miseries, has been given the name of Durgā. (Brahma Vaivarta Mahāpurāṇa 2:57:7-8)
What is the relationship between Lord Viṣṇu and Durgā?
In glory, lustre, beauty and virtues She is equated with Nārāyaṇa and She happens to be His sole energy. She is therefore called Nārāyaṇī. In the earlier times Lord Viṣṇu created the illusion (Māyā) and the entire universe was infatuated with the illusion. She was therefore called Viṣṇu-Māyā. She stands for the welfare of Śiva and is also beloved of Śiva, that is why She is called Śivā. In every aeon She pervades the universe as the Great Goddess of success. Being a chaste lady, possessing the best of nature, She is called Satī. As Lord Viṣṇu is eternal, similarly the Goddess Bhagavatī is also eternal; at the time of dissolution She merges into the body of Lord Kṛṣṇa. Therefore, everything from a straw to Brahmā is all artificial, but like the Lord, the Goddess Durgā is also truthful. As all the success of riches, the name present in all the aeons, the word Bhaga is added to the name of the same Goddess forming the word Bhagavatī. (Brahma Vaivarta Mahāpurāṇa 2:57:9–16)
The word Ambā is used for mother for adoration and worship. Therefore the one who is worshipped by all is called Ambikā, the devotees of Viṣṇu call Her Vaiṣṇavī since She happens to be the form of Viṣṇu, with whose energy the universe is created by Viṣṇu at the time of creation. She is of fair complexion, clad in yellow garments, unattached and spotless. Because She is the energy of the Great Soul, She is called Gaurī. She happens to be the spouse of Śiva, Lord Kṛṣṇa is also the teacher of all and She happens to be the illusion of Kṛṣṇa. She is therefore called Gaurī, because of the difference in date, difference in mountains, difference in kalpas and several other differences. She is thus called Parvatī. During the special occasion, the word parvan is used and since She happens to be the goddess of all the sacred occasions, She is called Parvatī. More so, because She happens to be the daughter of the mountain, appeared over a mountain and became the goddess of mountains and so She is called Parvatī. In the earlier times at the beginning of creation, Lord Kṛṣṇa adored Her in the Goloka in Rāsamaṇḍala of Vṛndāvana. Secondly Brahmā adored Her getting terrified for Madhu-kaiṭabha. Thirdly lord Śiva adored Her at the time of his war with Tripura. Fourthly in the earlier times Mahendra adored Her after he was cursed by Durvāsā as a result of which he lost all his riches and glory. Having fallen from glory, he adored the Goddess with great devotion. Since then, She was popularly adored by all the sages, the siddhas, the gods and the Ṛṣis everywhere. О sage, in the earlier times, She appeared from the lustre of all the gods under the name of Durgā. All the gods presented Her with their weapons and ornaments. She killed the Durga and other demons and bestowed the kingship of word to the gods. (Brahma Vaivarta Mahāpurāṇa 2:57:20-34)