Why do modern-day liberals (for example) generally consider it okay to say "I think everyone should be happy" without offering an explanation, but not okay to say "I think I should be free to keep slaves", regardless of the explanation offered? In an earlier age, the second statement might have been considered acceptable, while the first one would have required an explanation.
In general, people accept their favorite intuitions as given and require people to justify any intuitions which contradict those. If people have strongly left-wing intuitions, they tend to consider right-wing intuitions arbitrary and unacceptable, while considering left-wing intuitions so obvious as to not need any explanation. And vice versa.
Of course, you will notice that in some cultures specific moral intuitions tend to dominate, while other intuitions dominate in other cultures. People tend to pick up the moral intuitions of their environment: some claims go so strongly against the prevailing moral intuitions of my social environment that if I were to even hypothetically raise the possibility of them being correct, I would be loudly condemned and feel bad for even thinking that way. (Related: Paul Graham's What you can't say.) "Culture" here is to be understood as being considerably more fine-grained than just "the culture in Finland" or the "culture in India" - there are countless of subcultures even within a single country.
Social psychologists distinguish between two kinds of moral rules: ones which people consider absolute, and ones which people consider to be social conventions. For example, if a group of people all bullied and picked on one of them, this would usually be considered wrong, even if everyone in the group (including the bullied person) thought it was okay. But if there's a rule that you should wear a specific kind of clothing while at work, then it's considered okay not to wear those clothes if you get special permission from your boss, or if you switch to another job without that rule.
The funny thing is that many people don't realize that the distinction of which is which is by itself a moral intuition which varies from people to people, and from culture to culture. Jonathan Haidt writes in The Righteous Mind: Why Good People Are Divided by Politics and Religion of his finding that while the upper classes in both Brazil and USA were likely to find violations of harmless taboos to be violations of social convention, lower classes in both countries were more likely to find them violations of absolute moral codes. At the time, moral psychology had mistakenly thought that "moving on" to a conception of right and wrong that was only grounded in concrete harms would be the way that children's morality naturally develops, and that children discover morality by themselves instead of learning it from others.
So moral psychologists had mistakenly been thinking about some moral intuitions as absolute instead of relative. But we can hardly blame them, for it's common to fail to notice that the distinction between "social convention" and "moral fact" is variable. Sometimes this is probably done for purpose, for rhetorical reasons - it's a much more convincing speech if you can appeal to ultimate moral truths rather than to social conventions. But just as often people simply don't seem to realize the distinction.
(Note to international readers: I have been corrupted by the American blogosphere and literature, and will therefore be using "liberal" and "conservative" mostly to denote their American meanings. I apologize profusely to my European readers for this terrible misuse of language and for not using the correct terminology like God intended it to be used.)
For example, social conservatives sometimes complain that liberals are pushing their morality on them, by requiring things such as not condemning homosexuality. To liberals, this is obviously absurd - nobody is saying that the conservatives should be gay, people are just saying that people shouldn’t be denied equal rights simply because of their sexual orientation. From the liberal point of view, it is the conservatives who are pushing their beliefs on others, not vice versa.
But let's contrast "oppressing gays" to "banning polluting factories". Few liberals would be willing to accept the claim that if somebody wants to build a factory that causes a lot of harm to the environment, he should be allowed to do so, and to ban him from doing it would be to push the liberal ideals on the factory-owner. They might, however, protest that to prevent them from banning the factory would be pushing (e.g.) pro-capitalism ideals on them. So, in other words:
Conservatives want to prevent people from being gay. They think that this just means upholding morality. They think that if somebody wants to prevent them from doing so, that somebody is pushing their own ideals on them.
Liberals want to prevent people from polluting their environment. They think that this just means upholding morality. They think that if somebody wants to prevent them from doing so, that somebody is pushing their own ideals on them.
Now my liberal readers (do I even have any socially conservative readers?) will no doubt be rushing to point out the differences in these two examples. Most obviously the fact that pollution hurts other people than just the factory owner, like people on their nearby summer cottages who like seeing nature in a pristine and pure state, so it's justified to do something about it. But conservatives might also argue that openly gay behavior encourages being openly gay, and that this hurts those in nearby suburbs who like seeing people act properly, so it's justified to do something about it.
It's easy to say that "anything that doesn't harm others should be allowed", but it's much harder to rigorously define harm, and liberals and conservatives differ in when they think it's okay to cause somebody else harm. And even this is probably conceding too much to the liberal point of view, as it accepts a position where the morality of an act is judged primarily in the form of the harms it causes. Some conservatives would be likely to argue that homosexuality just is wrong, the way that killing somebody just is wrong.
My point isn't that we should accept the conservative argument. Of course we should reject it - my liberal moral intuitions say so. But we can't in all honestly claim an objective moral high ground. If we are to be honest to ourselves, we will accept that yes, we are pushing our moral beliefs on them - just as they are pushing their moral beliefs on us. And we will hope that our moral beliefs win.
Here's another example of "failing to notice the subjectivity of what counts as social convention". Many people are annoyed by aggressive vegetarians, who think anyone who eats meat is a bad person, or by religious people who are actively trying to convert others. People often say that it's fine to be vegetarian or religious if that's what you like, but you shouldn't push your ideology to others and require them to act the same.
Compare this to saying that it's fine to refuse to send Jews to concentration camps, or to let people die in horrible ways when they could have been saved, but you shouldn't push your ideology to others and require them to act the same. I expect that would sound absurd to most of us. But if you accept a certain vegetarian point of view, then killing animals for food is exactly equivalent to the Holocaust. And if you accept a certain religious view saying that unconverted people will go to Hell for an eternity, then not trying to convert them is even worse than letting people die in horrible ways. To say that these groups shouldn't push their morality to others is to already push your own ideology - which says that decisions about what to eat and what to believe are just social conventions, while decisions about whether to kill humans and save lives are moral facts - on them.
So what use is there in debating morality, if we have so divergent moral intuitions? In some cases, people have such widely differing intuitions that there is no point. In other cases, their intuitions are similar enough that they can find common ground, and in that case discussion can be useful. Intuitions can clearly be affected by words, and sometimes people do shift their intuitions as a result of having debated them. But this usually requires appealing to, or at least starting out from, some moral intuition that they already accept. There are inferential distances involved in moral claims, just as there are inferential distances involved in factual claims.
So what about the cases when the distance is too large, when the gap simply cannot be bridged? Well in those cases, we will simply have to fight to keep pushing our own moral intuitions to as many people as possible, and hope that they will end up having more influence than the unacceptable intuitions. Many liberals probably don't want to admit to themselves that this is what we should do, in order to beat the conservatives - it goes so badly against the liberal rhetoric. It would be much nicer to pretend that we are simply letting everyone live the way they want to, and that we are fighting to defend everyone's right for that.
But it would be more honest to admit that we actually want to let everyone live the way they want to, as long as they don't things we consider "really wrong", such as discriminating against gays. And that in this regard we're no different from the conservatives, who would likewise let everyone live the way they wanted to, as long as they don't do things the conservatives consider "really wrong".
Of course, whether or not you'll want to be that honest depends on what your moral intuitions have to say about honesty.