There are two basic kinds of statements that can be made, positive statements, such as "The sky is blue", purely descriptive, without any value judgement or goal or preference implied, and normative statements, such as "The sky should be blue," which are prescriptive, based on values or goals, but only describing an ideal or preferable state. The relationship between positive and normative can also be expressed as "is" and "ought", "fact" and "value", "descriptive" and "prescriptive", or "idea" and "ideal". Note in each pairing that the first word relates the state of the actual world, and the second relates to a state the world could be in that would be desirable.
There are cases where normative statements will pretend to be positive statements. These usually relate to ideals of "good" and "bad" or "evil". Grammatically, "sodomy is evil" looks like an is statement, but this is one of the most naked normative expressions possible. That's because normative statements aren't about the word "is" or "ought", but about value judgements. "Sodomy is evil" is a value judgement just as "You shouldn't commit sodomy" is a value judgement. "Helping others is good" is just as much a value judgement as "You should help others."
Then there are cases where, rather than "good" and "bad", one might say "right" and "wrong". This can be more difficult because we use "right" and "wrong" both in the prescriptive sense, synonymous with "good" and "evil", but also in a descriptive sense, synonymous with "correct" and "incorrect". Sometimes people will use the terms "right" and "wrong" specifically to hide the prescriptive aspects of what they say.
There are even murkier cases, such as "It is right that a man has his own house." or even "I have a right to a house." In the first, "right" is being used in the sense of right and wrong (as opposed to correct and incorrect, which are unambiguously descriptive rather than judging), and so it's not terribly complicated to see that this statement expresses a value. In the second sentence, "my right" is treated as an object, not as a value judgement, but if we look into what is meant, it is clear that any "right to a house" is not describing his relationship to a house, but prescribing his relationship to a house, that he should have one.
Very often these two words are used interchangeably, but some people use one or the other to mean specific things. The way I use the terms here is mostly interchangeably, because the differences aren't relevant to what I primarily need to say about them. In general though, "ethics" connotes picking the best thing to do out of a range of options, especially in a specific context, e.g. "journalistic ethics", while "morality" connotes deciding whether a particular action is good or bad, without worrying so much about which one is the best. If you are reading a work on ethics, you'll want to check how that particular author uses the terms "ethics" and "morality" to make sure you understand them correctly.
There is a well-established problem in ethics called the is-ought problem. The basic problem is that you can't validly derive any normative (i.e. prescriptive) conclusions from purely positive (i.e. descriptive) premises. Every valid logical conclusion that has normative content must have a normative premise. If that normative premise is in turn a conclusion of some other valid logic, then it must have a normative premise. If it is not a conclusion from other logic, then it is simply a naked value judgement. (Just to make the list exhaustive, if it is a conclusion of invalid logic, we generally don't care about the conclusion anymore.)
This means that all normative expressions are based ultimately on a naked value judgement. All kinds of different techniques are used to obscure the naked value judgement, but using the terms "right" and "wrong" confusingly is a popular one. A naked value judgement is the point from which you cannot derive back any further, and logic fails to rationalize it. If one disagrees with the naked value judgement, then all of the logical conclusions that follow from it are as easily dismissed and rejected as its weakest naked value judgement. And because all values are subjective, these may all be dismissed with no reason or justification needed, just as they can be asserted with no reason or justification needed.
When you introduce a normative premise into your system of logic, you are effectively defining an axiom. (Some readers may be under the impression that axioms are necessarily true; this is not the case, an axiom is a premise that delimits a system of logic.) This is the starting point of logic which assumes that all of its premises are true, and in cases where the axioms are not true, it is understood that the conclusions of that system of logic cannot be safely considered true. For example, the axioms of geometry in flat 2D space are different than those of geometry on the surface of a sphere, just because you prove that a triangle with three right angles in flat space is impossible doesn't mean the conclusion is valid on a sphere. Doing this in ethics allows you to reason about ethics, but it makes your conclusions relative to the normative axiom, just as the geometric conclusions are relative to a kind of 2D space.
If I say that killing is wrong, and begin to make logical deductions from this premise, all of my conclusions are true only relative to a system where killing is itself wrong. My conclusions will not all be true in a system where the axiom is "murder is wrong" rather than "killing is wrong". The truth of my conclusions is relative to the truth of my axioms (and the validity of my logic). The truth of my conclusions is not absolute, because my axioms are not absolute.
Because all normative expressions rely ultimately on these naked value judgements, no normative expressions can be absolutely true or false. They are all true or false only relative to the truth of their naked value judgement, which is true or false differently for different people who may or may not hold these values, or may prioritize their values differently.
This doesn't mean that there is no role in ethics for logic. It just means that the conclusions are not "absolute" in the way they might be when purely descriptive premises are used. Within these systems of logic, you can still reveal contradictions and falsify conclusions relative to the premises. This can be done with as much logical precision as it can outside of these systems of logic. This can still reveal insights into how we can best meet our varied and competing values as we pursue those values, and for some kinds of values which bring their own forces of tension with them, such as "equality" or "fairness", we can potentially falsify nearly every case where people might be treated unequally or unfairly and converge on some kind of understanding of equilibrium in these systems.
But even values that bring in their own opposing tensions are no more absolute than the simpler ones. All ethics are conditional to values, and all values are subjective, even our most treasured values.