I want to know who are Aśvinī-kumāra brothers? What are their names and what is the role of Aśvins or Ashwini Kumaras in our mythology? Are they demigods?
According to Shrimada Bhagavatam, Canto 8, Chapter 13, Asvini brothers are the son of Lord Surya and Vadava.
SB 8.13.8 — O King, I have previously described [in the Sixth Canto] the two daughters of Viśvakarmā, named Saṁjñā and Chāyā, who were the first two wives of Vivasvān.
SB 8.13.9 — It is said that the sun-god had a third wife, named Vaḍavā. Of the three wives, the wife named Saṁjñā had three children — Yama, Yamī and Śrāddhadeva. Now let me describe the children of Chāyā.
SB 8.13.10 — Chāyā had a son named Sāvarṇi and a daughter named Tapatī, who later became the wife of King Saṁvaraṇa. Chāyā’s third child is known as Śanaiścara [Saturn]. Vaḍavā gave birth to two sons, namely the Aśvinī brothers.
They are the physicians of Devas. It is mentioned in Shrimada Bhagavatam, Canto 9, Chapter 3.
SB 9.3.11 — Thereafter, some time having passed, the Aśvinī-kumāra brothers, the heavenly physicians, happened to come to Cyavana Muni’s āśrama. After offering them respectful obeisances, Cyavana Muni requested them to give him youthful life, for they were able to do so.
According to Vishnu Purana, Book 3, Chapter 2, Asvini brothers are the son of Lord Surya and Sanjna (instead of Vadava).
Sanjná, the daughter of Viśwakarman, was the wife of the sun, and bore him three children, the Manu (Vaivaswata), Yama, and the goddess Yamí (or the Yamuná river). Unable to endure the fervours of her lord, Sanjná gave him Chháyá as his handmaid, and repaired to the forests to practise devout exercises. The sun, supposing Chháyá to be his wife Sanjná, begot by her three other children, Śanaiśchara (Saturn), another Manu (Sávarńi), and a daughter Tapatí (the Tapti river). Chháyá, upon one occasion, being offended with Yama, the son of Sanjná, denounced an imprecation upon him, and thereby revealed to Yama and to the sun that she was not in truth Sanjná, the mother of the former. Being further informed by Chháyá that his wife had gone to the wilderness, the sun beheld her by the eye of meditation engaged in austerities, in the figure of a mare (in the region of Uttara Kuru). Metamorphosing himself into a horse, he rejoined his wife, and begot three other children, the two Áswins and Revanta, and then brought Sanjná back to his own dwelling.
Aśvinī Kumāras are the divine physicians. They are the twin sons of Surya and Saṁjñā born when they are in form of horses. Hence they are called Aśvinī Kumāra.
Their names are Dasra and Nasatya. They are always referred to as Aśvinī Kumāras.
Their birth is said in several puranas but Padma Purana Srishti Khanda has them in detail.
Saṁjñā, the daughter of Vishwakarma married Surya and had three children Yama, Yami and Vaivasvata manu (current manu). Unable to bear the lustre of Surya, Saṁjñā made a lady Chāyā of her and said:
To her, who stood before her saying; "what should I do ?" she said : "O beautiful Chaya., you serve my husband, nourish my children with motherly affection" Saying "All right" she of a righteous vow, went to god (Vivasvan) for sensual enjoyment. [41-42]
Then the lord taking her to be Samjna gave birth to Savarni Manu and Tapati from her. She continued to live like Saṁjñā and was partial towards his own son Savarni manu. Vaivasvata manu and Yama didn't tolerate it. Out of anger, he lifted his right foot onto her. She pronounced a curse.
Chaya too, cursed Yama: "This one foot will be full of worms and pus, and blood will ooze from it." 
Insulted by this curse, Yama told his father: "O god, mother has angrily cursed me without any reason. Due to child-like nature, I raised my foot a little. O lord, even though warded off by Manu, she gave me a curse. [48-49]
Having known that his wife had left him, Surya went to Vishwakarma to know the whereabouts of her.
Then Tvastr said these kind words to him : "O remover of darkness, not being able to bear your severe lustre, she came to me here, I, fearing you, kept her off, O lord of the sky. 'Since you have come to me here without your intention being known (by your husband), please do not enter my house'. [58-60]
Thus addressed, she the blameless one, quickly went to the Maru country and taking the form of a mare, lived on the earth. 
Vishwakarma took some lustre out of Surya and made his appearance delightful to the world with thousand rays but this time without his feet. As tvashtar removed the feet of the surya, no one should worship the imnage of Surya with feet. Then Surya took the form of a horse and approached her.
Then the love-striken lord of gods going to the earth in the guise of a horse and endowed with great lustre, had oral coitus with her; and the mind of Samjna, afflicted by fear, became agitated. [69-70]
Suspecting him to be a stranger, she threw out (his semen) through her nostrils. We have heard that from that semen Asvins were born. 
They became known as Dasra (wonder-workers) and Nasatya (truthful) (due to their being born) from the nostrils. 
Full of joy, he went to the heaven with his wife in an aeroplane. Savarnya Manu also is even now practising penance on Meru. 
Role of them in Mythology:
They are the fathers of vanara twins Mainda and Dvivida in Ramayana.
From Balakanda sarga 17,
Rūpa draviṇa saṃpannau aśvinau rūpasaṃmatau |
maindam ca dvividam ca eva janayāmāsatuḥ svayam || 1-17-14
The Ashwin twin gods whose wealth is their beautiful appearance have personally procreated two vanara-s namely Mainda and Dvivida, blessing them with beautiful appearance.[1-17-14]
In Mahabharata, Nakula and Sahadeva are born with the amsha of these deities.
Thus addressed by her lord, Kunti readily yielded, and said unto Madri, 'Think thou, without loss of time, of some celestial, and thou shall certainly obtain from him a child like unto him.' Reflecting for a few moments. Madri thought of the twin Aswins, who coming unto her with speed begat upon her two sons that were twins named Nakula and Sahadeva, unrivalled on earth for personal beauty. And as soon as they were born, an incorporeal voice said, 'In energy and beauty these twins shall transcend even the twin Aswins themselves.' Indeed possessed of great energy and beauty, they illumined the whole region. Mahabharata [Sambhava Parva SECTION CXXIV]
There are many other stories related to them like Chyavana Maharshi restoring their youth. They are the ones who recommended Agni to burn up the Khandava vana which was full of medicinal herbs when Agni was suffering from stomach ailment caused by consuming ghee from long fire sacrifices.
Are they Demigods?
If you mean, Devatas, yes, they are Devatas. They are extolled in Rig Veda in several places.