a quasi-coherent sheaf
On a scheme the structure sheaf O_X plays the role of 'functions'; a sheaf of O_X-modules is then the geometric version of a module over a ring — think of it as a bundle of vector spaces whose sections you can multiply by functions. But not every sheaf of modules deserves the name 'algebraic': you want the ones that, locally, come from honest modules over the coordinate rings. A quasi-coherent sheaf is exactly such a sheaf — one that is locally presentable by generators and relations as a cokernel of free modules — and it is the correct class of sheaves to do algebraic geometry with.
Precisely, on an affine scheme Spec R every R-module M produces a sheaf M-tilde whose sections over the basic open D(f) are the localized module M_f and whose stalk at a prime p is M_p; this gives an equivalence between R-modules and quasi-coherent sheaves on Spec R, the module-level version of the ring/affine-scheme anti-equivalence. On a general scheme X, an O_X-module F is quasi-coherent if X has an affine open cover Spec R_i on each of which F restricts to M_i-tilde for some R_i-module M_i. Equivalently, F is quasi-coherent if it is locally a cokernel of a (possibly infinite) map of free O_X-modules: locally there is an exact sequence O_X^{(I)} -> O_X^{(J)} -> F -> 0. The quasi-coherent sheaves form an abelian category closed under kernels, cokernels, extensions, tensor products, and pullback — exactly the operations algebraic geometry needs, which mere O_X-modules do not always respect.
Quasi-coherent sheaves are the basic objects on which sheaf cohomology of schemes is built, and they package ideal sheaves (defining closed subschemes), the modules of differentials, and all the standard constructions. The defining feature is the link to algebra: a quasi-coherent sheaf is determined by compatible module data on affine charts, so questions about it reduce to commutative algebra. Two honest cautions. First, 'quasi-coherent' is genuinely weaker than 'coherent': the module M may be infinitely generated, so quasi-coherent sheaves can be enormous (the sheaf associated to an infinite direct sum is quasi-coherent but not coherent). Second, not every O_X-module is quasi-coherent — for example a skyscraper sheaf supported at a non-closed point, or certain hand-built sheaves, can fail the local-module condition; quasi-coherence is a real restriction that ties the sheaf to the algebra of the structure sheaf rather than letting it be an arbitrary topological gadget.
On Spec R the ideal I of R gives the quasi-coherent ideal sheaf I-tilde, whose sections over D(f) are I_f; the quotient O_X / I-tilde is the structure sheaf of the closed subscheme Spec(R/I). For R = k[x, y] and I = (y), this is the quasi-coherent sheaf cutting out the x-axis. Everything reduces to the modules I and R/I and their localizations.
Ideal sheaves are quasi-coherent: the module I on Spec R sheafifies to cut out a closed subscheme.
Quasi-coherent is weaker than coherent: the local modules may be infinitely generated, so these sheaves can be huge. And not every O_X-module is quasi-coherent — quasi-coherence ties the sheaf to genuine module data on affine charts.