A Vaishnava-based answer ahead. Lord Brahma in the Devi Kavacham, a part of Devi Mahatmyam of Markandeya Purana, says to Sage Markandeya:
प्रथमं शैलपुत्री च द्वितीयं ब्रह्मचारिणी ।
तृतीयं चन्द्रघण्टेति कूष्माण्डेति चतुर्थकम् ॥ ३ ॥
पञ्चमं स्कन्दमातेति षष्ठं कात्यायनीति च ।|
सप्तमं कालरात्रीति महागौरीति चाष्टमम् ॥ ४ ॥
नवमं सिद्धिदात्री च नवदुर्गाः प्रकीर्तिताः ।
उक्तान्येतानि नामानि ब्रह्मणैव महात्मना ॥ ५ ॥
[Devi Kavacham, Devi Mahatmya, Markandeya Purana]
So the Navadurgas are basically Shailaputri, Brahmachaarini, Chandraghanta, Kushmaanda, Skandamaata, Kaatyaayani, Kaalaraatri, Mahagauri, and Siddhidaatri. These are the names that are spoken by the high-souled Lord Brahma. These nine forms of Durgadevi are the forms of Umadevi’s Durga roopa. Both Lakshmidevi and Umadevi have their own Durga roopa. Lakshmi devi’s Durga roopa does the samhaara of everybody in the universe, whereas Umadevi’s Durga roopa, known as Sharvaani, does the samhaara of other lokas up to Svarloka, along with Lord Shiva, who is known as Sharva, because He does samhaara. As in the Lakshmi Tantra, Lakshmidevi says that She is Durgadevi; in the same way, in the Gauri Tantra too, Umadevi says that She is Durgadevi. So, both have their Durga roopas.
Lakshmidevi’s Durga roopa is Mahakali and Mahalakshmi, in which She killed Madhu-Kaitabha and Mahishasura. Because in those incidents, the devatas pray to Her as ‘Vishnumaaya’, which means Vishnu’s maaya. Plus, Krishna in the Bhagavad Gita says:
दैवी ह्येषा गुणमयी मम माया दुरत्यया ।
मामेव ये प्रपद्यन्ते मायामेतां तरन्ति ते ॥ १४ ॥
[Bhagavad Gita, Chapter 7, Verse 14]
Here, Krishna uses the term मम माया दुरत्यया, which means, ‘My Maaya’. The माया to whom He is referring as His here is Lakshmidevi, who is Durgadevi, the Mahamaaya. She also has the name Maayadevi; that’s why Vishnu is known as Maayapati. So, this Durga is Vishnumaya Durgadevi, who is being worshipped in the Devi Mahatmya. She is not married to anyone and is a Kanyaka form of Lakshmidevi. Therefore, an unmarried woman (Kanya) has the presence of Durgadevi within them. Even in the Uttara Charita of Shumbha-Nishumbha Vadha, the devatas pray:
या देवी सर्वभूतेषु विष्णुमायेति शब्दिता ।
नमस्तस्यै नमस्तस्यै नमस्तस्यै नमो नमः ॥ १४ ॥
So, in that story itself, when Umadevi asks the devatas whom they are praying to, a Devi emerges from Umadevi’s body cells, who is Mahasarasvati. This is Umadevi’s Durga roopa. This Umadevi’s Durga roopa has Vishnumaya Durga within Her. When the Devi came out of Umadevi’s body cells, She said, “They are worshipping me”, and She is the Vishnumaya Durga who is present there, within Umadevi’s body. Umadevi’s Durgaroopa further expanded into the Navadurgas, which we see now. These names, like Shailaputri, Brahmachaarini, Chandraghanta, Skandamaata, etc., are for Umadevi.
Why is Durgadevi worshipped in these nine roopas? The first name is Shailaputri. Shailaputri means ‘The one who is the daughter of the mountains.’ Umadevi is the daughter of Himavanta; that’s why She is known as Shailaputri. Second, She is known as Brahmachaarini. She is ever-devoted to Lord Shiva and never sees any other purusha except Lord Shiva. She is a great Pativrataa; that’s why She is known as Brahmachaarini. Third is Chandraghanta. She is known as Chandraghanta because, in Her Durga roopa, She has Chandra as Her ornament on Her head.
Fourth one is Kushmanda. Vishnumaya Durgadevi is the form of Lakshmidevi. During creation, Lord Vishnu kept the whole Brahmanda in Lakshmidevi's womb to give happiness and joy to Lakshmidevi. So, from Her and Lord Vishnu came the whole Brahmanda; therefore, She, along with Lord Vishnu, is the cosmic creator. This same Vishnumaya Durgadevi is present within Umadevi’s Durgaroopa; therefore, Umadevi is also known as Kushmaanda. Fifth is Skandamaata. Since She is the mother of Skanda/Kartikeya/Subrahmanya/Murugan/Shanmukha, She is known as Skandamaata.
Sixth is the Katyayani roopa of Umadevi. This was the roopa that the Gopikas worshipped in Vrindavana to get Krishna as their husband. When everyone was frustrated by the nuisance of Mahishasura, a light came from the devatas and merged into one roopa of Devi named Mahalakshmi. This Mahalakshmi then went to Katyayana Maharshi and accepted his worship, and came out of him in another roopa. That’s why She is also known as Katyayani. This Katyayani roopi Devi is the same Vishnumaaya Durga present in Umadevi’s Durgaroopa; therefore, Umadevi’s Durga roopa is also known as Katyayani.
Seventh is Kaalaraatri. Kaalaraatri is the Kaali who came out of Mahasarasvati’s (Umadevi’s Durga roopa) forehead to kill Chanda and Munda. This is the fiercest form of Devi, in which She is grabbing every demon by the legs, hair, hands, tongues, faces, heads, etc., and eating them up. Finally, She killed Chanda and Munda and even Raktabeejasura. For the asuras, who are against Dharma, She is like death to them. Eighth is Mahagauri. Umadevi is known as Gauri because Her body is golden in color. She is known as Mahagauri because She has brilliant golden-colored skin.
Ninth is the Siddhidaatri roopa. This is also a form of Umadevi, through which She is the provider of all siddhis and knowledge. She has great knowledge about Bhagavan, plus Umadevi is a great devotee of Bhagavan Vishnu. She does Shaastrapaatha to Indra and other devatas, and hence is their guru. Lord Shiva is the niyaamaka for the bodies of Indra and other devatas, plus is the astra-shastra guru for Indraadi devatas. These are the names of the Navadurgas which were spoken by Lord Brahma Himself to Sage Markandeya.
Why should we worship these Navadurgas? Because, Lord Brahma further says:
अग्निना दह्यमानस्तु शत्रुमध्ये गतो रणे।
विषमे दुर्गमे चैव भयार्ताः शरणं गताः॥6॥
न तेषां जायते किञ्चिदशुभं रणसङ्कटे।
नापदं तस्य पश्यामि शोकदुःखभयं न हि॥7॥
यैस्तु भक्त्या स्मृता नूनं तेषां वृद्धिः प्रजायते।
ये त्वां स्मरन्ति देवेशि रक्षसे तान्न संशयः॥8॥
Those who are surrounded by raging fire, those who find themselves trapped in the midst of enemies on a battlefield, or those who are stuck in highly treacherous, inaccessible places (durgame)—if, overwhelmed by absolute fear, they take refuge in You...
...no inauspiciousness or harm shall ever befall them in the crises of war, nor will they ever face calamity. They shall be entirely free from grief, sorrow, and fear.
Those who remember You with deep devotion (bhaktyā) will certainly experience continuous growth and prosperity (vṛddhiḥ). O Goddess of the Divinities (Deveshi)! You protect those who remember You—of this, there is absolutely no doubt.
This Vishnumaya Durgadevi Herself is dependent on Lord Vishnu for Her powers, since He is the origin of Her powers, as She Herself says in the Ambhruni Sukta: Mama yonirapsvantah samudre
Hope this helps!
Jay SriRam!
ShriDurgaantargata SriKrishnarpanamastu!