James Fieser divides the study of ethics into three subject areas:
The field of ethics (or moral philosophy) involves systematizing, defending, and recommending concepts of right and wrong behavior. Philosophers today usually divide ethical theories into three general subject areas: metaethics, normative ethics, and applied ethics. Metaethics investigates where our ethical principles come from, and what they mean. Are they merely social inventions? Do they involve more than expressions of our individual emotions? Metaethical answers to these questions focus on the issues of universal truths, the will of God, the role of reason in ethical judgments, and the meaning of ethical terms themselves. Normative ethics takes on a more practical task, which is to arrive at moral standards that regulate right and wrong conduct. This may involve articulating the good habits that we should acquire, the duties that we should follow, or the consequences of our behavior on others. Finally, applied ethics involves examining specific controversial issues, such as abortion, infanticide, animal rights, environmental concerns, homosexuality, capital punishment, or nuclear war.
These three subject areas can be further divided.
He focuses on two issues of metaethics:
Two issues, though, are prominent: (1) metaphysical issues concerning whether morality exists independently of humans, and (2) psychological issues concerning the underlying mental basis of our moral judgments and conduct.
He breaks normative ethics into three strategies:
The key assumption in normative ethics is that there is only one ultimate criterion of moral conduct, whether it is a single rule or a set of principles. Three strategies will be noted here: (1) virtue theories, (2) duty theories, and (3) consequentialist theories.
Applied ethics includes specific issues. The problem here is to distinguish when an issue is an applied ethical issue from social policy issues. He notes two criteria he claims are necessary for an issue to be an applied ethics issue:
First, the issue needs to be controversial in the sense that there are significant groups of people both for and against the issue at hand.
The second requirement for an issue to be an applied ethical issue is that it must be a distinctly moral issue.
Let's consider the question: So, what are the classifications of these ethics?
Fieser's way to classify what people do when they write about ethics is to consider if the subject area is metaethics, normative ethics or applied ethics. From there one can continue dividing the subject areas as needed.
James Fieser, "Ethics" Internet Encyclopedia of Philosophy https://www.iep.utm.edu/ethics/