A conceit is an unconventional metaphor that compares two things in illogical or surprising ways as opposed to more conventional metaphors comparing things that are more apparently similar, like sunrise being like awakening garden blossoms. As a side note, the term "conceit" is often restricted to the metaphysical conceits, which compare the abstract with the physical sensory, that are used by metaphysical poets of the 1600s (17th century), but it is not correct to say conceits were only developed by the metaphysicalists. "Conceit" comes from the Latin for "concept": a poetic conceit is a complex metaphoric concept.
From the Latin term for “concept,” a poetic conceit is an often unconventional, logically complex, or surprising metaphor whose delights are more intellectual than sensual. (PoetryFoundation.org)
The conceit around which Sidney's sonnet cycle (also, though less eloquently, called a sonnet sequence) Astrophil and Stella is built is a conventional sonnet conceit initiated by Italian poet Francesco Petrarca, whom we commonly refer to as Petrarch. Petrarch developed the sonnet form and the convention of sonnet conceits. The sonnet form was later borrowed and developed by English poets, most notably, Howard (Earl of Surrey), Wyatt, Spenser, Sidney, and Shakespeare.
The conceit that governs Sidney's sonnet cycles is the complex and illogical one that is predicated on (based on) the illogical assumption and belief that a person's emotions, thoughts and desires can be "read" as one reads written words. This conceit has become such an ingrained part of English metaphor that it no longer seems surprising to us.
That she, deare Shee, might take som pleasure of my paine,
Pleasure might cause her reade, reading might make her know,
Knowledge might pittie winne, and pity grace obtaine,... (I)
... euen thus: in Stellaes face I reed
What Loue and Beautie be; then all my deed
But copying is, what in her Nature writes. (III)
The "reading" conceit might be said to be built within a primary conceit illogically comparing love to war ["Loue gaue the wound, which, while I breathe, will bleede (II)] where, in the midst of love's battle, each person's feelings and thoughts may be read in their faces, gestures and behaviors in the same way that Astrophil's outpouring of feelings and thoughts may literally be read in the written words of Sidney's sonnets.
The relationship between love and war had been a common Italian poetic conceit ever since the time of Petrarch in the 14th century (MagnificatBaroque.com)