The fourteen "abodes" of knowledge are: the four vedas; the six Angas or limbs of the Vedas; Mimamsa, Nyaya, the Puranas and Dharmasastra. You must have seen at least references to the Vedas and the six Angas. The Tamil work Tevaram says: "Vedamodarangamayinanai". According to this devotional work Isvara is the form of the four Vedas and the six Angas.
The fourteen dharma-pramanas (authorities of dharma) are called "caturdasa-vidya". The well-known poetic work 'Naisadham' mentions that Nala was conversant with these fourteen branches of learning. The poet (Sriharsa ) plays on the word "caturdasa": he says that "Nala accorded caturdasa to the caturdas-vidya", meaning he gave the fourteen branches of learning four dasas: reading, understanding what is read, living according to the teachings contained in what is read, and making others also live in accordance with them.
Caturdasatvam Krtavan kutah svayam
Na vedmi vidyasu caturdasasvapi
--Naisadham, 1. 4
All religious knowledge is encompassed by these fourteen branches of learning.
There are yet four more vidyas. If you add to the fourteen already mentioned, you will have eighteen vidyas - astadasa-vidya which are all-inclusive. Of them, the fourteen already mentioned are directly concerned with dharma. The remaining four - Ayurveda, Arthasastra, Dhanurveda and Gandharvaveda - do not directly deal with dharma. They are not dharmasthanas (abodes of dharma) but they qualify to be vidyasthanas(abodes of knowledge). The first fourteen, as already mentioned, are both dharmasthanas and vidyasthanas (abodes of dharma as well as abodes of knowledge).
The dharmasthanas and vidyasthanas are together commonly known as the sastras. The word "sastra" means an order or commandment. We speak of a royal "sasana", meaning a royal "edict". There is a chapter in the Mahabaharaaata in which Bhisma expounds the ordinances of dharma to Yudhisthira and it is called "Anusasana-parva". Aiyanar is called "Sasta" because he keeps the hosts of Siva under his control (through his orders ). Works on sastras incorporate the ordinances that are calculated to keep us disciplined and ensure that we tread the right path.
While all the fourteen sastras are basic and authoritative texts, the Vedas are their crown. Just as Buddhism, Zoroastrianism (Zarathustrianism), Christianity and Islam have the Tripitaka, the Zend-Avesta, the Bible and the Qur'an respectively as their scriptures, we have the Vedas as our prime scripture.
Of the fourteen branches of learning the first four (the four Vedas) form the basis for the subsequent ten. Together they constitute the complete corpus of sastras on which our religion is founded.
For a general background, please see here