Many Buddhist texts and teachings make a distinction between absolute and relative truth. Please explain the difference and give some examples.
In Theravada, as the Wikipedia article states, there are types of truth: conventional truth (sammuti-sacca) and ultimate truth (paramattha-sacca).
These two types of truth are not exactly the same as relative and absolute truths. Rather than focusing on the universality or specificity of the scope of the truth, they focus on the intrinsicness of the truth - i.e., whether an object agrees with a truth by its inherent nature, or whether it is simply said to agree with it by external convention.
An example of conventional, extrinsic truth is in a fist. When you close your hand, a fist appears, almost by magic. When you open your hand again, the fist disappears. There was never anything intrinsic about the object that made the fist appear; only convention and recognition gave rise to the truth or falsehood of the arising of a fist.
An example of ultimate, intrinsic truth is in the pain felt when a fist connects at high velocity with a face. The fist and the face are only conventional descriptions of the reality, but the pain is intrinsic to the experience and thus said to be ultimately real. Also ultimately real are the vision of the fist coming towards the face, the feeling of the hardness of the fist and the softness of the face, and the sound of the fist connecting with the face.
In brief, conventional reality refers to entities, while ultimate reality refers to the building blocks that make up reality (i.e. experience). So, the entities of fists and faces are conventional, but the sights, sounds, and feelings associated with the experience are ultimate reality.
The importance of this doctrine is in establishing a proper paradigm for understanding reality. Because conventional truths are based on mental formations, they cannot aid in an understanding of reality. A permanent, satisfying, controllable convention can be conceived of, but it will never exist in ultimate reality. By delineating that which exists in ultimate reality, one can limit one's observation to those objects that will provide an understanding of how reality works.
For example, focusing on the concept of food as you eat, you will never come to see the nature of reality as arising and ceasing incessantly, because you are focused on something that is constant (but not really existing) - i.e. the food. As a result, you will easily conceive of it as good or bad, me and mine. If you focus on the experience of tasting and feeling as you chew the food, you will notice that you don't actually taste or feel the food always, your mind flitting between tasting, feeling, liking, disliking, thinking, etc., and you will come to understand the nature of reality, which in turn will allow you to let go of your attachments to good an bad experiences as you see that there is nothing truly desirable or undesirable about them.
This is based on the teaching of the Buddha while he was alive.
Relative and absolute truth, I believe, are more a part of the Mahayana claim that the teachings of the Buddha during his life were only meant to be taken in the context of the audience's limited ability to understand higher truths. Such teachings, it is claimed, can therefore be disregarded or at least superseded by later teachings that were kept secret until a time when beings could understand the higher truths they contained.
So, an example of a relative truth would be the teaching of dependent origination and the four noble truths; ultimate truth would be seeing that even those teachings are emptiness, and therefore meaningless. The point is that in order to understand the ultimate truth of emptiness, one must first realize the conventional truths as stepping stones.
I'm sorry if I haven't done the Mahayana view justice, but I don't subscribe to it myself; there is nothing in the original teachings of the Buddha to suggest that they are only relatively true, or that there is anything absolute beyond them. But, I guess I'm biased :)
As @Yuttadhammo pointed out, relative and absolute truths are a Mahayana construct. In Pali Canon, Buddha does not declare such duality (all he does is caution his students against taking his metaphors literally).
From Mahayana standpoint, the phenomenal world, consisting of subject and its objects as well as relationships between them, is relative truth. Any description of reality is necessarily in the realm of relative truth as it depends on arbitrary axioms posited against each other. The ultimate truth ("what really IS") is therefore ineffable, but can be tapped in practice through deconstructing all (relative) constructs.
The difference between Relative and Ultimate truths becomes important for understanding Mahayana concept of Enlightenment:
From Mahayana perspective, the Path to Nirvana an individual walks belongs firmly in Relative Truth. However, because ultimately no separate "individual" exists apart from the interacting mental-and-physical factors, there is no entity that achieves Nirvana. Instead, when Three Marks of existence are fully realized as Emptiness of Interbeing, the notion of Nirvana is no longer applicable and Self-Existing Enlightenment can finally be realized.
Reality is an interpretation we make. Our everyday world, with its "I" and "others", birth and death is one such interpretation. The world of Buddhist practitioners, with its skandhas, samskaras, dharmas, and nirvana is another interpretation, adopted temporarily as a stepping stone from the first. Regardless of where one is on the Path, Ultimate Truth remains the ground reality underlying all worlds.
Two truths is not just Mahayana development, it's Abhidharma (Abhidhamma) development, and exists in Theravada too (check this (pdf) excellent article of prof. Karunadasa). But "relative and absolute" is really misnomer, properly said it's conventional and ultimate truths (skt. samvriti & paramartha satya). Their distinction is like between naïve language (convention) and scientific analytical discourse. When language is used in its common sense it's conventional, and when it express Buddhist teaching it's truth. When reality as it is, is analysed into dharmas and expressed in special dharma-discourse (adhidharma), it is ultimate truth.
Typical example is: when Buddhist teaching talks about beings it's conventional truth, when it talk about skandhas it's ultimate truth.
The centerpiece of his Madhyamaka or ‘‘middle way’’ philosophy is the thesis that everything is empty. This thesis has a profound consequence. Ultimate truths are those about ultimate reality. But since everything is empty, there is no ultimate reality. There are, therefore, no ultimate truths. We can get at the same conclusion another way. To express anything in language is to express truth that depends on language, and so this cannot be an expression of the way things are ultimately. All truths, then, are merely conventional.
This is what Jay Garfield and Graham Priest calls "Nagarjuna’s first limit contradiction." More on this in Nagarjuna and the limits og thought" http://www.thezensite.com/ZenEssays/Nagarjuna/NagarjunaTheLimitsOfThought.pdf
Narjuna on the relation between the two truths: The notion of ultimate truth (reality) (paramārtha-satya, literally ‘‘truth of the highest meaning,’’ or ‘‘truth of the highest object’’) can be defined negatively as the way things are, considered independently of convention, or positively as the way things are, when understood by a fully enlightened being who does not mistake what is really conventional for something that belongs to the very nature of things.
What is ultimate truth/reality, according to Nāgarjuna? To understand this, we have to understand the notion of emptiness, which for Nāgarjuna is emphatically not nonexistence but, rather, interdependent existence. For something to have an essence (Tibetan: rang bzhin; Sanskrit: svabhava) is for it to be what it is, in and ofitself, independently of all other things. (This entails, incidentally, that things that are essentially so are eternally so; for if they started to be, or ceased to be, then their so being would depend on other things, such as time.)
To be empty is precisely to have no essence, in this sense. The most important ultimate truth, according to Nagarjuna, is that everything is empty. Much of the Mulamadhyamakakarika consists, in fact, of an extended set of arguments to the effect that everything that one might take to be an essence is, in fact, not so—that everything is empty of essence and of independent identity.
Dependent arising, relative truth, is emptiness and emptiness is the ultimate truth.
This realtion can be expressed with a slogan, namely "The Ultimate Truth Is That There Is No Ultimate Truth" (Mark Siderits )
I'm a Vajrayana practitioner and this is how the two truths were explained to me in very simplified terms.
On a relative level, which describes our every day experience, dualistic and conceptual thinking prevails and one feels separated from others. On the absolute level, however, there is no duality, and object, subject and action are the parts of the same totality. On the absolute level everything is empty of intrinsic existence. This should not be confused with nihilistic thinking that things are hollow or not-existent - they do appear but they are interdependent and have no lasting properties. A great Indian Yogi Saraha phrased that in the following way:
People who think that things are real are as stupid as a cow;
People who think things are not real are even more stupid.
Understanding emptiness and thus the absolute truth is difficult and that is why the relative truth is employed to learn about the nature of mind. On the absolute level, there is exactly the same Buddhahood in a sacred temple as in a bathroom, but because of our habitual likes and dislikes, our meditation is deeper in a temple. A highly realised Lama has the same Buddha Nature as a criminal but on a relative level there is much more benefit to trust the Lama.
Once the emptiness is understood, the veils of ignorance disappear and a great joy is being discharged. Understanding the nature of the mind is indeed the lasting happiness and there is no greater joy than that. The following short poem nicely captures that:
Since everything is but an illusion,
Perfect in being what it is,
Having nothing to do with good or bad,
Acceptance or rejection,
One might as well burst out laughing!
Longchen Rabjam (1308-1363)
The concept of relative/absolute truth in early Buddhism is pretty straight-forward. Ven. Bodhi's note from his "Middle length Discourses" says:
The Buddha has a twofold teaching(2 levels of truth)-a conventional teaching/sammutidesana expressed in terms of persons, beings, women, men,etc and an ultimate teaching/paramatthadesana expressed solely in terms that possess ultimate ontological validity, such as aggregates, elements, sense bases, impermanent, suffering, not self,etc. The Buddha expounds his teaching thru whichever approach is best suited to enable the hearer to penetrate the meaning, dispel delusion, and achieve distinction..
For further info., also see Ven. Nyanatiloka's definition of paramattha from his "Manual of Buddhist Terms and Doctrines" here
The separator, between them, are causes (to arising), and imagination.
Ultimate truths are not imagination, because they are effects that arise by their own causes. Except nibbana, ultimate truth, is the opposite of the other causes and effects, because nibbana is the cause of finshing of all causes and effects.
Conventional truths are imagination. They never arise and vanish. They just become to be the objects of consciousness, never arise.
Consciousness imagine conventional truth objects by some ultimate truths and some conventional truths, that it had known before. So the conventional truths never arise in real situation. But ultimate truths arise in real situation by their own causes, because they are effects, not just imagination.
The example already available in the ancient canon:
The 1st-7th and first half of 8th chapters of abhidhammatthasangaha describe Ultimate truths.
The last half of 8th chapter of abhidhammatthasangaha describe Consciousness truths.
Absolute truth (pure truth) is free of its opposite which is falsehood. Whereas relative truth is always attached to falsehood like the two sides of a coin. Relative truth is the always a result (an effect) of a cause; where as, absolute truth is never a result (effect) of anything and this is the reason why there is no path to Nirvana! Because defining a path means absolute truth (Nirvana) suddenly becomes the result (effect) of taking that particular path.