The term "formalism" is normally used by post-modernist or post-structuralist critics to refer to structuralist or New Critical approaches to literature. It is often used in a reductionist manner, and refers to the practice of what I. A. Richards called "practical criticism" and is pedagogically termed "close reading" of looking carefully, word by word, at a work (usually a poem) and analysing its stylistic and formal characteristics (meter, figures of speech, patterns of imagery, etc.) with looking at such matters as historical or cultural background. Under this definition, deconstruction, unlike new historicism or gender theory, is actually a type of formalist criticism, despite deconstructionists' habits of critiquing and distancing themselves from New Criticism.
Deconstruction is a post-structuralist movement associated with Jacques Derrida, that searches for hidden assumptions or binary oppositions in texts and attempts to invert or collapse them.
One way to look at formalist criticism is to see it as a modern version of the kind of rhetorical criticism that has a long and valuable tradition in the history of western literature. Rhetorical criticism was very prominent, for instance, during the age of William Shakespeare. Goerge Puttenham'sThe Arte of English Poesie, for instance, is just one of many such treatises written during the so-called English Renaissance. Puttenham and other writers like him are interested in the nitty-gritty details of poetic phrasing to a degree that is still astonishing. Thus, one standard charge against formalism -- that it is ahistorical and anachronistic, an invention of the twentieth century that has been imposed on literature of the past -- is difficult to accept.
It is true that formalism was revived and more thoroughly articulated in the 1920s and 30s by the so-called "New Critics," who merely argued that literature should be read as literature-- that is, as writing that calls attention to itself as writing. For a magisterial defense of this position, see the book titled Theory of Literature, by Rene Wellek and Austin Warren, which remains impressive decades after it was first published.
One main objection to formalism by later schools, including deconstruction, is the assertion that formalists look for (and inevitably find) "unity" in works of literature and that this unity is imposed rather than genuinely present in "the work itself" (a favorite New Critical phrase). In contrast, deconstruction (as the very term suggests) looks for ways in which any work inevitably lacks unity and is riven by contradictions. Deconstructors also try to collapse neat distinctions, such as the distinction between "fiction" and "history," showing how such terms cannot be easily distinguished from one another.
Most of the standard charges against formalism -- such as the claim that it tends to ignore historical and social contexts -- are not entirely fair, and in any case it is easy enough to imagine a kind of formalism that avoids such charges. Thus, so-called "historical formalism" has been one response to these kinds of allegations. In general, formalism seems to be making a bit of a come-back these days after having been attacked for half a century. People once again seem increasingly interested in reading literature as language that calls attention to itself as language.