Anscombe was a student of Wittgenstein's and is most famous for translating his Philosophical Investigations, and when Bro pitched this topic to me, he described her as the transition from Wittgenstein to Alasdair MacIntyre. This is puzzling, as Wittgenstein talks largely about language, whereas MacIntyre is all about virtue ethics.
In her essay "Modern Moral Philosophy" (1958) she claims that there's something wrong with modern ethics' use of "should." Of course, in ordinary language we run into many occasions where "should" is appropriate, as in "In chess, the bishops should only move diagonally" or "In chess, you should anticipate your opponents' moves." These kinds of "shoulds" are of course not considered part of morality, but Anscombe also says claims like "It's unjust to convict an innocent man" are also part of familiar, everyday parlance and do not require some kind of overarching philosophical ethics to be agreed upon. A plant needs water; people need love; if I borrow money from you, then I owe it back to you. All of these are likewise evaluative claims that she finds unproblematic. (And, to make the Wittgenstein connection, could be looked at as something like language games, though she doesn't use that term.)
Where modern ethics makes its mistake is then to pretend that these various bits of value can be summed up into a world of raw good and evil, where "should" is used not just in a given context as in the above but as an overall requirement on our behavior. She points to Aristotle's ethics as not making this mistake. For Aristotle, different organisms have their "good," their "virtues," but no obligation arises from these. Yes, since you're a person, you should be smart and just and prudent and eat foods that are good for you and get enough rest. But given those facts about human constitution, are you then obliged to do and be those things? Is there a moral law that requires you to do them?
Anscombe says that we only think in terms of moral laws and obligations because Christianity was dominant in our culture for so long.
It is as if the notion "criminal" were to remain when criminal law and criminal courts had been abolished and forgotten. A Hume discovering this situation might conclude that there was a special sentiment, expressed by "criminal," which alone gave the word its sense. So Hume discovered the situation which the notion "obligation" survived, and the notion "ought" was invested with that peculiar for having which it is said to be used in a "moral" sense, but in which the belief in divine law had long since been abandoned: for it was substantially given up among Protestants at the time of the Reformation. The situation, if I am right, was the interesting one of the survival of a concept outside the framework of thought that made it a really intelligible one.
Hume rightly discovered, she said, that supposed evaluative facts (e.g. that old chestnut of rationalist ethics: "It is fitting to respond to generosity with gratitude") don't actually motivate us to do anything just by us contemplating them. So he decided that morality must then be a type of sentiment. Anscombe says no: we have aims, we pursue activities, we have intentions, and all of that puts us in a position where we follow advice like that of Aristotle mentioned above, or the advice about playing chess, or running a just society. We don't have to associate these aims we have with "desires" in the sense of an inner impression as Hume thought, and again, these various evaluative bits do not and need not sum up to a special moral sentiment or anything like that. She likewise disses Hume's supposedly unbridgeable "is-ought gap"; she thinks evaluation is already built into these localized practices, and it is unilluminating to the point of distorting to try to tease apart the purely descriptive element and the normative element from our various terms, practices, and judgments.
Perhaps surprisingly, given the above, one of the sources of her objection to modern moral philosophy is her absolutist stance about certain prohibitions, such as killing the innocent. Shortly before writing this essay, she argued against Cambridge's award of an honorary degree to Harry Truman on the grounds that his atomic bombing at the end of WWII was unconscionable. He used the deaths of innocents as a means to acquire Japan's unconditional surrender, and this is right out for her. According to a utilitarian, Truman was justified if the good outweighed the harm, but for Anscombe, this means that all prohibitions are subject to compromise, which is a sign of utter corruption: you have no principles if this is the case.
She also wants to distinguish between cases like the above where killing innocents was vilely used as a means to an end and a case where, e.g. one's intent is to take out a munitions site (killing only soldiers), but there is a foreseeable risk that this will also result in the deaths of innocents. In such a case, you don't use those innocents as your means. This is the distinction between intended results (Truman intended to kill innocents) and merely foreseen but not intended results (bombing the munitions plant will kill innocents, but that was not the point of the plan).
She gives a clearer case of this "double effect": You're in a position where if you don't commit some ethical violation, you'll be put in prison, in which case your family will go hungry. A consequentialist might say "do the ethical violation if it wouldn't cause anything worse than you abandoning your family in that way; if you choose to abstain, you'll be responsible for them being left out in the cold." Anscombe thinks that we need to distinguish again between the intended (you are responsible for whether you do the unethical deed or not) and the merely foreseen (you are not responsible for this foreseen event that you do not intend).
How does this link up with her argument in "Modern Moral Philosophy"? Well, some people think that she's just trying to argue indirectly that we really should return to a divine command approach to morality. If you instead take her at her word, then this view of what you might call localized normativities (standards internal to a language game only, not generalizable into "the good" overall) can still be applied directly to these kinds of dilemmas. One has to decide how to act in a tough situation, and if you're a consequentialist, you just sum up all the good and bad anticipated consequences, choose the best (or least bad) option, and you've done the right thing. For Anscombe, I think, even if you do decide that circumstances necessitate that you commit some otherwise unjust act (I kill one innocent to save 1000), you might be understood and forgiven, but that still wouldn't make the act a just one in this circumstance: the local normative judgments stay accurate and don't get summed up into a command that you get praised if you obey.
We also read her essay "War and Murder" from 1961, an explicitly Catholic essay where she discusses legitimate state power, pacifism, and the double effect. Download it here.
Intention is a remarkable work in itself, giving a systematic analysis of what an intention is (hint: it's not a feeling you have when you do something) and how to describe intentional actions (Do you describe Truman's action as "winning the war" or "killing innocents" or "bombing Hiroshima" or "giving an order to his generals" or what?). If you're a fan of Wittgenstein or want to see a really good example of analytic philosophy, I'd recommend that book over the two essays.