This technical term provides a good example of the mis-named, the loosely-defined, the nameless, and the only-belatedly-named in Middle English poetics. It's a word sometimes flung about by students when close-reading: any syntactical break (i.e. anywhere a line of poetry has a division between grammatical phrases or clauses, and especially if it's near the middle of the line) gets called a caesura without too much thought. It's one of those technical terms that gives faux expertise: spotting something easy (i.e. a syntactic break) and labeling it with a word that makes it seem as if metre is being analyzed. Joseph A Dane is usefully strict about how relevant the word is for Chaucer's metre in this article (the answer is not very!).
The word itself (deriving from the verb 'to cut') comes from Latin metrical theory. Here it describes the situation which occurs when the gap between words does not align with the break between metrical feet (i.e. where the end of a word occurs in the middle of the foot rather than at the end). Some types of metre require or encourage a word-break which is not aligned with a foot-break (that is, a caesura, as opposed to diaeresis, where the word-break and the break between feet coincide) at a certain point in the line.
As often in poetics, theorisers borrow a word from another language's technical terms and use it with a related but different meaning (just as Greek terms like iamb and dactyl have been adapted). Caesura is used somewhat loosely by scholars of Middle English verse to refer to pauses in a line of verse, whether metrical, syntactical, or rhetorical (i.e. a pause added for emphasis or dramatic effect). Calling a syntactical or rhetorical pause a caesura muddies the waters as to what is a metrical feature (i.e. what plays a part in the vast majority of lines) and what is a feature of the rhythm of an individual line.
Several different types of Middle English verse regularly divide their lines up into two half-lines (also sometimes called hemistichs), sometimes of equal length and with the same metrical structure, sometimes of different lengths and sometimes with different metrical structures. When the caesura is a structural feature of the vast majority of lines, this creates certain expectations for the reader and constraints for the writer. The line's syntax has to be constructed by the poet to have a break at a certain point. A word won't be split across the gap. Yet norms are not unbreakable rules: the patterns of sense and syntax work with and against the pattern of the caesura. Macklin Smith has a great essay showing how Langland varies the impact of the caesura in the line and at times uses syntax to resist the underlying pattern of the mid-line break.
According to Nicholas Myklebust's detailed PhD study, Chaucer's decasyllabic line does have syntactic and rhetorical pauses but doesn't have regular metrical caesurae. Chaucer's lines may have syntactical and/or rhetorical pauses, but this is not a feature of every line. Myklebust's PhD also shows that whereas Chaucer and Hoccleve did not have a regular metrical caesura, Lydgate divides each of his lines into two half-lines. Lydgate draws on French practice in vers de dix, which (in some varieties at least) had a caesura at a fixed point after the fourth syllable in the line.
As Myklebust explains brilliantly, the word before the mid-line break is rather like a line ending (i.e. a place where self-imposed constraints and self-allowed licences might apply). Licences and constraints are common at the line end: most obviously rhyme, but also in 'classical' alliterative poetry, where lines must end with an unstressed final syllable, whilst Chaucerian decasyllabics allow an unstressed syllable after the final beat. Similar things go on at the ending of the first half-line. The word before might have to rhyme with the word at the end of the line (creating what is often called medial rhyme). The poet might impose the constraint that the syllable before the caesura should be a monosyllable or a strongly stressed syllable. Poets might also allow themselves licences to do things at the caesura which they might not do elsewhere in the line. Lydgate, as Myklebust explains, uses the licence of the caesura to add or delete a syllable. He was not a poor metrist who couldn't count to ten, but rather a poet who divided his decasyllable line into two parts and allowed himself metrical licences at the caesura.
Despite its presence in several different types of Middle English verse, regular pauses built into the structure of the line do not seem to be named or discussed much before the middle of the sixteenth century. It is understood as one element of the self-imposed constraints of poetic form, but it is neither named nor theorised in relation to medieval English verse (at least as far as current research suggests). The earliest usage of the word itself seems to be by Archbishop Matthew Parker, when giving advice to the lay reader on the reading through of English psalms in preparation for singing (1567). In giving this advice, Parker is still thinking of psalmody rather than prosody, how to read these English translations in imitation of Latin psalm-singing. He writes:
Observe the trayne: the ceasure marke, / To rest with note in close: / Rythme-dogrell playne: as dogs do barke, / ye make it els to lose.