Cleft constructions split a clause into two parts to foreground and focus a constituent: 'It is Mary who left' (it-cleft) or 'What I want is coffee' (pseudo-cleft). Clefts obligatorily focus the clefted phrase and differ in which phrase types and syntactic positions permit clefting.
From your work on information structure, you know that sentences package old and new information — they mark a topic (what the sentence is about) and a focus (the new or contrastive information the speaker highlights). Most information packaging is handled by prosody in speech or word-order choices in writing. Cleft constructions offer a more explicit grammatical strategy: they split a single clause into two parts, isolating the focused element in a way that makes its prominence structurally unavoidable rather than leaving it to intonation.
The most common type is the it-cleft: "It is Mary who left early." The neutral version of this sentence is "Mary left early." The cleft extracts "Mary," places it after the copula, and wraps the rest of the original clause into a relative clause. The effect is to obligatorily focus "Mary" — the construction conveys: among the people who might have left early, it is specifically Mary who did. Notice that the cleft also does something your information structure knowledge should recognize: it presupposes that someone left early (background, carried in the relative clause) and asserts who (the focused element). The presupposition is embedded in the structure.
The second major type is the pseudo-cleft (or *wh*-cleft): "What I want is coffee." Here a *wh*-clause occupies subject position and the focused element appears as the predicate following the copula. Pseudo-clefts are sometimes called specificational sentences — the *wh*-clause sets up a variable (what I want = ?) and the focus specifies its value (coffee). They handle long or complex focused elements more gracefully than it-clefts: "What the committee decided after lengthy deliberation was to table the motion indefinitely" is natural; fitting that focus into an it-cleft would be awkward.
Cleft constructions carry a pragmatic signature: they presuppose their background clause and are therefore infelicitous when that presupposition fails. "It was John who arrived late" is odd if nobody arrived late — the construction presupposes that someone did. This presuppositional behavior distinguishes clefts from simple focus marking through stress. Understanding clefts means understanding not just which element they focus, but what they take for granted — the background that must already be established for focusing a particular element to be a coherent communicative act.