Simone de Beauvoir was an existentialist, and lifelong partner of Sartre. Her blurb loosely refers to The Ethics of Ambiguity, which is one of the major works in existentialist ethics. She describes how the ambiguity of every situation leads to anxiety in the subject, because he is forced to act without knowing what the right thing to do is, or even perhaps what he wants to do. Any attempt to categorize the world or morality into concrete, definable rules is bound to fail, both in practice (we will still be left with an ambiguity of how to apply the rules) and in theory (objective rules cannot supersede free subjective decisions, since morally arises from subjectivity). To overcome this anxiety and become an authentic moral agent, you must accept the ambiguity and act anyway.
Sartre's blurb refers to the fact that he practices phenomenology, which is the study of the structures of experience. Via Husserl, he was then 'bracketing' (i.e. putting aside) metaphysical questions, such as the existence of the outside world or other minds, in order to study experience itself. If I had more space I might have said he wields 'Husserl's amulet of phenomenology', since that concept is more tied to Husserl. His dialog with Kant is concerning his ideas that you are in 'bad faith' (lying to yourself, essentially), which you deny your freedom by pretending that a hard and fast rule must be followed, such as a previous oath.
Kant's blurb refers to his categorical imperative, which says that an act is not moral if it results in a contradiction if universalized. Lying is textbook example for Kant, if everyone lied, then no one would believe anything anyone said, and lying itself would be pointless; for Kant this means that lying is always wrong. Kant was then challenged, and asked if there were a murderer at the door, would it be acceptable to lie to him about he whereabouts of his victim? Kant stuck to his guns and said that you still can't lie to the murder, lying is always wrong. The rest of his dialog is along similar lines, such as following through with his duties, and fulfilling oaths, which are also necessary in Kant's philosophy.
Foucault was known primarily for his 'genealogical' accounts of different forces in our society, such as how we punish criminals, treat the mentally ill, or our ideas on sexuality. A genealogical account is essentially a history of the ideas which lead to our current ideas on the topic, usually, for Foucault - through the lens of power structures. The comic most closely resembles Madness and Civilization, which specifically mentions Lepers, and how the mad took over their ideological place in early modern Europe as leprosy became less common. Foucault was also an anarchist, and fought for groups' right to self governance and freedom from oppression from more powerful groups.
Derrida was a post structuralist (as is Foucault), and is best known for his critiques of language and meaning. Structuralism essentially claims that we understand the meaning of words by how the relate to other words, in opposition. So for example, we might understand what 'evil' means as it opposes 'good', but without the opposition of good we wouldn't be able to understand 'evil' by itself, it's meaning comes from its contrast to other words. Post structuralism, as I understand it (not very well), is an internal, structuralist critique of structuralism itself. He basically claims that even the structure of the words is not adequate to understand the meaning of a word, since the meaning of the word always depends on its context, both current and historical, and that context is never stable. Therefore it is impossible for words to have fixed meanings or be understood completely.