The earliest, and perhaps simplest, kinds of feminist criticism were concerned with recovering ignored works by women. In "A Room of One's Own" (1929), for instance, Virginia Woolf famously writes about the problems that would have been faced by Shakespeare's (imaginary) sister, and Woolf's works include many readings of women authors who had not been taken seriously. This kind of feminist criticism continues today, turning up many once-important figures who've been excluded from the canon. There's also been plenty of continued squabbling over whether these "rediscovered" writers belong in the canon.
Other comparatively simple kinds of feminist criticism include biographical and historical studies of women writers, women readers, or women's issues in literature, and examinations of the way women are depicted in literature. Some of this can of course be quite sophisticated in its use of close reading and its exploration of history; this is, if I can make a very rough generalization, the dominant mode of American and British feminist criticism.
To continue the rough generalization, French feminist critics tend to be more indebted to various kinds of theory, and I can do no more than hint at the many varieties. Some feminist critics have drawn on psychoanalytical theory, especially that of Jacques Lacan. They look at the way children acquire a language structured around binary oppositions that give the privileged place to masculinity, and associate it with power, reason, activity, and so on; femininity is passive, irrational, and so on. Language itself, in their view, is "phallocentric."
Others, again especially in France, have suggested that there's room for a use of language that escapes phallocentricity: that there's an écriture féminine, "feminine writing," a kind of approach to language that does not impose on the world the rigid hierarchies associated with masculine writing. It is a kind of use of language that escapes the patriarchalism of the dominant discourse. For some feminist critics like Hélène Cixous and Luce Irigaray, écriture féminine is grounded in the female body and feminine sexuality; the term jouissance covers both female sexual pleasure and textual play. Note, though, that écriture féminine isn't strictly limited to women writers: men, too, can engage in "feminine writing."
Note, once again, that the division into French and Anglo-American "schools," while conventional, is only a rough guide. Plenty of Anglo-American feminists have drawn from French critics, and vice versa. And I should once again insist that this is a sadly superficial summary of a huge field.
Note: This guide is still in the early stages of development.
Three question marks mean I have to write more on the subject. Bear with me.