Nyaya is also called Tarka-sastra and its author is Gautama. Its main purpose is to establish by reasoning that the Karta or Creator of all this world is Parameswara. Indeed, it seeks to prove the existence of Isvara through inference. Reasoning thus has a major place in Nyaya.
Logic or reasoning is of course indispensable to any study. The Vedas make a statement and Mimamsa determines its meaning. Though we have faith in the Vedas, doubts arise in our minds regarding the meaning of scriptural passages. If these doubts are cleared through reasoning the message of the Vedas will be affirmed. When we construct the marriage pandal we test the strength of the bamboo or timber posts by trying to shake them. In the same way we must subject truths to proper tests so as to confirm them. All logical reasoning must be accepted but it must be firmly rooted in authority. Also, arguments must not be of a carping character, stemming from the urge to be merely contrary.
When Sankara was about to depart from this world his disciples requested him for a brief upadesa. It was then that he imparted his succinct teaching in the form of five stanzas which go by the name of Upadesa-Pancaka or Sopana-Pancaka. "Dustarkat suviramyatam - Srutimatastarko'nusandhiyatam", is a line from it. It means that you must give up the habit captious arguments and that in dealing with a question you must employ proper reasoning, duly respecting the views of the Vedas.
Without reason to guide us it is like roaming aimlesssly in the forest. But reason must be founded on authority. Nyaya finds the meaning of Vedic passages in this manner.
Kanada too created a Nyaya sastra: it is called Vaisesika. One object is distinguished from another on the basis of the special characteristics or "particularities" of the two. The name "Vaisesika" is derived from the fact that it inquires into such particularities. There is a good deal of science in this Nyaya sastra. Atmic matters like the individual self, the cosmos, Isvara, moksa or liberation are examined (in Vaisesika "moksa" is known by the name of "apavarga").
The Nyaya inquiry into truth is through the four pramanas or instruments of knowledge of "pratyaksa", "anumana", "upamana" and "sabda". Pratyaksa is direct perception, what is perceived by the eyes and the ears and so on. It is anumana or inference that is central to Nyaya. What is anumana? We see smoke rising from the summit of a distant mountain: we notice only the smoke, not the fire, which is concealed by the rock perhaps. But even if we do not see the fire we may infer that the forest has caught fire. This is anumana. Here the fire is called "sadhya" and the means by which we infer its presence is "sadhana", "linga" or "hetu".
In our Vedantic system we must reflect upon the teaching imparted by our guru. This is manana and it means going over an idea (in this case the instruction received from the teacher) again and again in the mind, making use of our own ability to reason. Here anumana is of help. Is it not through inference that we are able to know things that cannot otherwise be perceived? The individual self and the Paramatman are not directly perceived by our senses. Nor do we know the liberation of senses. Nor do we know the nature of liberation or how to attain it. We have to know such things by inference. Knowing an object on the basis of another known object is anumana. When we hear the roar of the thunder we know, by inference, that there are clouds [that the sky is overcast]
By performing Vedic works [let us take it] we have become pure within. We have also found a good teacher and we have faith in his instruction. But, if we happen to hear something different from what he tell us, doubts naturally arise in our minds. These doubts have to be cleared; they must be discussed and a decision arrived at. Here we must have recourse to a pramana (source or instrument of knowledge) like anumana or inference. Both Nyaya and Vaisesika conduct inquiries based on anumana.
For a general background, please see here