Concepts and conceptions
Underlying seemingly unterminable discussions, different interpretations of words can often be pinpointed as the cause of trouble. Understanding the difference between concepts and conceptions might help you to come closer to an understanding of the disagreement.
Roughly, a concept is a general idea that is largely used by many different people in more or less the same way. It is usually a word or a phrase, embedded in a certain grammar that tells how it is used. A conception is a further specification of that concept. Tentatively, you can see a concept as something that we are most unlikely to disagree about, while a conception, being a specific outlook on a concept, is something about which disagreement is more likely, or at least possible. Disagreement on conceptions is even possible between reasonable, rational, and intelligent persons. In particular, it is possible between philosophers (which is not to say that philosophers are typically reasonable, rational or intelligent, or at least not by merit of their being a philosopher).
Just to give an example, we could take a brief look at the concept of justice. The concept could be explained something like: 'a situation of justice is one in which all deliver their duties, and all receive their entitlements'. Who would disagree? (Indeed, I only talk about distributive justice here as the problem of distributing goods between members of a society; matters of just punishment [retributive justice] and justified warfare will be skipped over here.)
In order to find different conceptions of justice, we only need to look at the two different sides of the Atlantic Ocean. Especially in most Northern-European countries, the incumbent conception of justice specifies that a large part, sometimes even above 50%, of persons' income may be taken by the state. We call this taxation, and it is justified by its use to provide common facilities such as healthcare, education and collective retirement schemes. In the United States, on the contrary, tax rates are considerably lower. It is believed there that the state has no right to take away from its citizens what they have earned by hard work. Through American eyes, many European systems are plain robbery. Through European eyes, the American system fails to offer the basic provisions that the needy are entitled to.
Of course, these are very, very simplified explanations of both systems - caricatural, if you like. Moreover, on either side of the pond, people exist who disagree to the scheme entertained by their state. In fact there is no such thing as the American view or the European conception of justice. Yet, what matters is that both systems are systems of justice in their own right.They are internally consistent, and you will be able to find reasonable, rational and intelligent persons who support one of them.
What matters more, is that a dispute between representatives of either view cannot be settled by an appeal to the concept of justice - that would produce exactly the exchange of views without getting any closer, which you do indeed sometimes see between Americans and Europeans. From the recognition that they actually use different conceptions when using the same concept, it is only a small step to starting investigating those conceptions. What is desert? What is entitlement? What are basic needs and who is responsible for providing them? Asking these questions after the specific conceptions is often a way out of interminable disputes.
Some philosophers (including rational, reasonable and intelligent ones) prefer the distinction between formaland material definitions. The distinction largely maps onto the difference between conceptions (material definitions) and concepts (formal definitions). However, the word 'definitions' has the ring to them of being explicitly posited. Concepts and conceptions on the contrary, can quite comfortably used as referring to 'being used while leaving their meanings implicit'. As that is exactly part of the problem, I prefer the usage above.
Reference and further reading
I have largely taken the distinction between concepts and conceptions from Adam Swift, Political philosophy - a beginners guide for students and politicians, Polity Press, Cambridge, 2001, p. 11.
If you are fascinated by this matter, you may want to read W.B. Gallie, Essentially contested concepts, Proceedings of the Aristotelian Society, 1956.