I've heard the term thrown around, but I haven't gotten a good explanation of the core tenets of that particular branch of ethics. Can anyone explain, in plain English, what it is?
Two key articles to read to understand care ethics:
Basic answer: feminist care ethics is a project building on the work of Carol Gilligan and Nel Noddings. Gilligan was critical of Kohlberg's stages of moral development. Lawrence Kohlberg came up with six stages of moral development (Kohlberg is a social scientist rather than philosopher) such that the apex of moral development is the ability to consider the abstract notion of justice. Gilligan objected and suggested instead that the epitome of being ethical is to have relations of care with others. In Kohlberg's study, only men tended to achieve this highest stage -- hence the objection to making this the "highest" being identified with feminism.
A bit later, Nel Noddings wrote a book called Caring (1984) in which she tries to work out how to build an ethics around relations of caring distinguishing between the one-caring and the cared-for. There's also a distinction between moral care and the feeling of care. This was after the fact considered a "feminist" ethics.
Separately, there's some argument about what makes something "feminist" or not with care feminists representing one of the camps within contemporary feminism. The other two main camps are (1) those views that see the differences between men and women as minor and the project of modernity as identifying morality and (2) those views that think notions of self are bogus.
The idea behind the ethics of care is that ethics is not an isolated factor of an independent person. People are interdependent, and truly moral actions may most logically flow from feedback and empathic cycles rather than from internal principles or decisions. So the appropriate way to increase moral behavior is to form bonds of care between individuals and for everyone to be aware of the relationships that already exist, rather than to emphasize responsibility and consistency, as is the trend to date.
There is nothing wrong with living in a dependent state, in fact it is unavoidable. Men (as aspiring 'heads' of supposedly separable units of social order) have just been trained to reframe it out of their thinking. So 'mushy' or 'indecisive' ethics is seen as less responsible and of lower value. But it may simply be a more objective view of what kinds of real ethical processes are ultimately effective. And it may produce better results. We find that by conservative standards women have less mature ethics, but behave better, and this makes no sense if the common, more rigid, view of ethics is right.
This causes us to have a skewed definition of what responsibility is throughout our society, And this may cause us to overestimate those with various kinds of ethics that may seem stable, but are in fact quite fragile. (Not to be too catty -- but for example we have seen a stream of high-profile leaders on the Right (Denis Hastert, Jerry Sandusky...) who seem far too prone to take advantage of young men and boys. Their principles on such issues are firm and clear, so they seem ethically mature. But they evidently cannot abide by them without some context. And when their egos are inflated, they have made horrible choices.)
This also means that one should actively cultivate 'mushiness' of this sort, the recognition of interdependence and willingness to respond to the subjective sense of helping those about whom you care, so that you do not make isolated decisions that have impact on others and their actions which you might not notice or might disown by imagining you are more separate from them than you really are.