There is an objective truth. It's one truth that's the same for all people. This is the common sense view. It means there is one answer per question.
Sometimes people get confused because they ask an ambiguous question. They use the same words to mean two different things, and then they think objective truth means there has to be one answer for both meanings of the question. The important thing is the actual question, not the words used. If I say "What day of the week is it today?" there is only one true answer, which is based on what day I ask that on. If you ask the same question on another day, there is again only one true answer to your question. If you want, think of the question as including an implied, "given that today is [the current date]," to remove the ambiguity.
Relativism is "the doctrine that knowledge, truth and morality exist in relation to culture, society or historical context, and are not absolute". The primary appeal is that people have realized, for example, that "Is it rude to do X?" is a different question depending on the context, especially depending on what culture you live in. This is the same kind of issue as the day of the week above. Relativists are correct to insist that a lot of the ideas of our culture are not universal truths, even some that most people assume are universal truths.
However, relativism is incorrect because it says that all knowledge depends on the context. It's a bit like saying that all questions are ambiguous just because some are and because precision is difficult. Also, relativism is ambiguous about whether contextual knowledge is absolutely true within that context; many relativists object to the idea of any absolute, permanent, unitary truth. But why should the truth for a given context ever change? Relativism provides an argument that the context is important, but no argument that the truth can change if we keep the context constant.
We have some knowledge that doesn't depend on context, called universal knowledge. It's knowledge that equally well applies to all situations. For example, the laws of physics don't have exceptions depending on which culture you live in. Gravity is the same no matter what the historical context or what your religion is. Physics is general purpose. Similarly, water consists of two hydrogen atoms and one oxygen atom, everywhere and forever. And 2+2=4 even if the majority opinion in my society thinks otherwise. (Of course we still have to be careful with ambiguity. Two buckets of water, plus two more buckets of water, can make one large puddle.) Relativists also can get themselves into trouble by saying there is nothing really wrong with rape, it's just a matter of opinion.
In general, any piece of knowledge has some limited amount of applicability. It is correct in some situations, but has some exceptions. The more widely knowledge applies, the better. Universal knowledge is a goal of both science and philosophy. Although it's hard to come by, we have made progress in discovering it, and can continue to make more. For example, computers would not function if we hadn't made genuine scientific progress.
The other reason people deny there is an objective truth is because they are skeptical that we know the truth. I accept that we never know what the truth is for sure. We can be mistaken. But to be mistaken, there has to be an objective truth! The idea of a mistake is that there is a truth and we have it wrong. Denying there are true ideas also denies that there are false ideas and mistakes. As a consequence, it prevents us from finding and correcting our errors, because errors are deviations from the truth and they say there is no truth to deviate from. Because they prevent error correction, relativism and other denials of truth are irrational ways of thinking. Similarly, denying there is a truth involves denying we can make progress, because progress means getting closer to the truth and understanding more of the truth.
Part of the issue is the idea that knowledge is justified, true belief. It's easy to lapse into relativism with that conception of knowledge because that kind of knowledge is impossible to come by, so one might think that opinions are all we have. Justificationists pave the way for relativists by denying that imperfect knowledge is knowledge at all.
Here is an argument that objective truth must exist: Communication relies on there being an objective truth. When I say something, you hear it. We're in a shared world. What you hear isn't random, it has to do with what I said. It's not based on your whim or subjectivity. What you hear is a close approximation of what I actually said, because you seek the truth of what I said and it is there to be found. Communication is only possible when there is one single truth of what is being said for all the people communicating.
Acknowledgments: some ideas presented are modified from, or inspired by, ideas from Karl Popper, Ayn Rand, William Godwin, Edmund Burke, David Deutsch and Thomas Szasz.