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One of Nāgārjuna's main contributions to Buddhist though was the development of the doctrine of sunyata or emptiness. This was a reaction to the development in the doctrines of the Abhidharma by some of the early Buddhist schools (for instance the Sarvastivada). In those schools there developed the concept of things (dharmas) that had some form of independent existence i.e were not subject to dependent origination. I will quote wikipedia on this point
The Madhyamaka school [Nāgārjuna traditionally seen as founding this school] has been perhaps simplistically regarded as a reaction against the development of the Abhidharma, especially the Sarvāstivādin. In the Abhidharma, dharmas are characterized by defining traits (lakṣaṇa) or own-existence (svabhāva), whose ontological status is not dependent upon concepts.
Nāgārjuna therefore refuted this and insisted that all things are inherently empty and do not have an independent existence. This means everything not just sentient beings. Nāgārjuna is also linked to the Perfection of Wisdom sutras and the Heart Sutra is particularly good on this point (please note I'm using the translation that we use in the Triratna Buddhist Community - scroll halfway down this document for the full version).
So, in emptiness, no form, No feeling, thought, or choice, Nor is there consciousness. No eye, ear, nose, tongue, body, mind; No colour, sound, smell, taste, touch, Or what the mind takes hold of, Nor even act of sensing.
So nothing in the psychophysical organism
No ignorance or end of it , Nor all that comes of ignorance; No withering, no death, No end of them.
Or ignorance or death/withering (so no lakshanas maybe?)
Nor is there pain, or cause of pain, Or cease in pain, or noble path To lead from pain; Not even wisdom to attain! Attainment too is emptiness.
Or even no four noble truths or eightfold path. So not much left.
So hopefully this giving a flavour of the radical nature of emptiness.
Please note this isn't a complete answer. There are other areas of Nāgārjuna's philosophy that I haven't covered such as the two truths largely due to a complete absence of knowledge on my part.