Vyaasam Vasishtha naptaaram
Sakteh Poutra Makalmasham
Paraasaraatmajam vande
Suka Taatam Taponidhim
Meaning: I bow to Vyaasa, the great grandson of Vasishtha, the grandson of that pure sage Sakti, son of Parashara and to the father of the great taponidhi Suka!
It is interesting to see that Vyaasa is glorified for being the father of Sage Suka, who is described as the treasure of tapas, penance.

This sloka is the third in Sri Vishnu Sahasranama Stotram. Obviously it is added by someone other than sage Vyasa! The next sloka is another tribute to sage Vyaasa.

Vyaasaaya Vishnu Roopaaya Vyaasa Roopaaya Vishnave
Namovai Brahma Nidhaye Vaasishthaya Namo Namah||
Meaning: I bow to Vyaasa, who is the treasure house of Brahma Jnana, who is of the clan of Vasishtha (all descendents of Vasishtha are Vaasishthas), who is verily one with Vishnu. It is interesting to note that both Vyaasa and Vishnu are portrayed as dark colored. Because of his dark complexion and because he was born in a riverine island, he is also known as Krishna Dwaipaayana. He is also called Paaraasharya, because he is a descendent of Paraashara. Krishna means dark colored and apart from the son of Devaki, who is most famous with that name and who is the 8th Avatar of Vishnu (it is interesting to note that He was also the 8th son of Devaki), Draupadi, the daughter of Drupada and the wife of the five Paandavas, was also known as Krishna. At the same time, she was also known to be quite beautiful with a glow of her own. We will have more to say about her at the appropriate time.

Vyaasa is considered as one of the many Upa Avataaras of Vishnu. Veda Vyaasa is a post and there will be a different Vyaasa for every Maha Yuga. Son of Paraasara is the Vyaasa for this Maha Yuga. Aswatthama will be the Veda Vyaasa for the next Maha Yuga. We will learn more about him at the right time.

Birth of Sage Veda Vyaasa:
Once Sage Paraasara was trying to cross the river Yamuna and he found one boat, but instead of a boatman, a beautiful young girl was in the boat. She offered her respects to the sage and offered to take him across the river. As the boat reached an island in the middle of the wide river, due to Divine Will, desire arose in the mind of the sage and he asked her to fulfill his desire. She objected politely saying that it was broad day light, people are there around and are watching, she is the daughter of a fisherman and is not desirable and she is still a virgin and if she accedes to his request, no one else will marry her. The sage explained to her that such desire will not normally arise in his mind and that it has arisen is due to Divine will and he will take care of all her objections: By his power acquired due to great penance, he made the whole place dark (it is said that he created smoke or fog that made visibility zero), he gave her a boon that a beautiful perfume will be wafting from her body for a distance of one Yojana (8 miles) thus making her very special and desirable, and that her virginity will remain unaffected and she will in fact marry a king. Thus satisfied, she accepted his request and immediately as a result of their union, Vyaasa was born on the island in the river. He grew to his full height with full knowledge of the Vedas and other Shastras within no time and took leave of his mother with a promise to her that he will manifest himself before her whenever she remembers him. Paraasara too went on his way.
Another version of Vyasa’s birth:
It is said that there have been twenty-eight Vyasas before the present Vyasa-Krishna Dvaipayana-took his birth at the end of Dvapara Yuga. Krishna Dvaipayana was born of Parasara Rishi through the Matsyakanya-Satyavathi Devi-under some peculiar and wonderful circumstances. Parasara was a great Jnani and one of the supreme authorities on astrology and his book Parasara Hora is still a textbook on astrology. He has also written a Smriti known as Parasara Smriti which is held in such high esteem that it is quoted by our present-day writers on sociology and ethics. Parasara came to know that a child, conceived at a particular Ghatika or moment of time, would be born as the greatest man of the age, nay, as an Amsa of Lord Vishnu Himself. On that day, Parasara was travelling in a boat and he spoke to the boatman about the nearing of that auspicious time. The boatman had a daughter who was of age and awaiting marriage. He was impressed with the sanctity and greatness of the Rishi and offered his daughter in marriage to Parasara. Our Vyasa was born of this union and his birth is said to be due to the blessing of Lord Siva Himself who blessed the union of a sage with a Jnani of the highest order, although of a low caste.
(From http://www.hindupedia.com/eng/index.php?title=Maharshi_Vyasa )
Thus because he was born on a dweepa (island), he came to be known as Dwaipaayana. Because it was dark and he was also dark complexioned, he was known as Krishna Dwaipaayana. He was also called Paaraasharya, and Saatyavateya, son of Satyavathi. Because he did his penance etc. at Badari, he was also known as Baadaraayana. There is another version about his getting the name Badarayana:

The Brahma Sutra is attributed to Badarayana — which makes him the proponent of the crest-jewel of school of Hindu philosophy, i.e., Vedanta. As the island on which Vyasa was born is said to have been covered by Badara (Indian jujube) trees, he is known as Badarayana. Though traditionally, Vyasa is considered the Badarayana who wrote the Sutras, many historians think these were two different personalities. (Wikipedia)

Dhritarashtra, Pandu and Vidura are his sons to Ambika and Ambalika and their maid respectively to keep the Kuru dynasty going after the sudden death of Vicithraveerya, the step-brother of Gaangeya, also known as Bheeshma. Vyaasa also got another son Suka who became a great sage in his own right and made Vyaasa famous. We will take up that story in detail when we take up Suka (also known as Suka Deva, Suka Deva Goswami etc.).
His writing of Mahabharatha: When sage Veda Vyaasa was asked by Lord Brahma to write Maha Bharatha, he was looking for a suitable scribe and Brahmaji persuaded Lord Ganesha Himself to take up the task and complete it. Lord Ganesha accepted it on one condition that once Vyasa starts the recitation, he should not stop it (His ghantam – scriber should not stop). Vyasa agreed and put a counter-condition that Ganesha should understand the text being recited before writing it down.
His bestowing the power of Television to Sanjaya:
Sage Veda Vyaasa wrote all the Puraanaas (18 main and 18 upa puraanaas), Mahabharatha (Bhagawad Gita and Sri Vishnu Sahasranama Stotra are part of Mahabharatha)and Brahma Sutras. He played an important role in Mahabharatha. When the war was becoming inevitable, Dhritarashtra being born blind was anxious to know what is happening but was finding it difficult. At that time Sage Veda Vyaasa visited him and at his request, gave a special power to Dhritarashtra’s charioteer Sanjaya that enabled him to see and hear and remember what is going on at the battle field while being physically present at Hastinapura, thus enabling him to give a running commentary to his king! Modern technology about 6000 years ago!

Veda Vyaasa and Basar:
Sage Veda Vyaasa is immortalized in many ways including the present town of Basar (which is slang for Vyaasapuri), near Nizamabad in Andhra Pradesh, where a temple for Goddess Saraswati was established by the sage. The temple of Saraswati at Basar is a major pilgrimage center especially for doing Aksharaabhyasam (initiating ceremony into the process of learning the alphabet) for children.

Veda Vyaasa and the Shankara Tradition:
It is known that Sri Shankara wrote his commentary to Brahma Sutras, which were written by Vyaasa. A few centuries later, the head of Sringeri Saarada Peetham, Sri Vidyaranya Bahrathi started on a pilgrimage and reached Kashi, Sri Veda Vyasa, sent him to Badarikashrama where the great sage himself initiated him into Sri Vidya. Thus it is clear that Sage Veda Vyaasa himself was a great exponent of Sri Vidya.