the stalk of a sheaf
A sheaf assigns data to whole open sets, but often you want to zoom all the way in to a single point p and ask: what does the sheaf look like in an arbitrarily small neighborhood of p? The trouble is there is no single smallest open set around p to evaluate the sheaf on. The stalk is the device that answers this by taking a limit over all shrinking neighborhoods at once — it packages together every section defined near p, while declaring two sections the same if they already agree on some (possibly tiny) common neighborhood.
Precisely, the stalk of a sheaf F at a point p, written F_p, is the direct (filtered colimit) limit of F(U) over all open sets U containing p, ordered by reverse inclusion, with the restriction maps as transition maps. An element of F_p is called a germ at p: it is represented by a pair (U, s) with p in U and s in F(U), and two pairs (U, s), (V, t) represent the same germ if there is a smaller open W containing p, inside both U and V, on which s and t restrict to the same section. So the stalk forgets how big a neighborhood a section is defined on and remembers only its infinitesimally-near-p behavior. For the sheaf of continuous functions the stalk at p is the ring of germs of continuous functions at p; for holomorphic functions it is the ring of convergent power series; for the structure sheaf of a scheme it is a local ring.
Stalks matter because they are where sheaf theory becomes local and computable. A morphism of sheaves is an isomorphism if and only if it is an isomorphism on every stalk — global behavior is detected pointwise through stalks — and sheafification is built by gluing the stalks back together. The single most important fact in scheme theory is that the stalks of the structure sheaf are local rings (one maximal ideal each), which is what makes a scheme a LOCALLY ringed space. A caution: stalks see only arbitrarily-small-neighborhood data, so two genuinely different sheaves can have isomorphic stalks at every point yet differ globally if the isomorphisms do not patch into a single sheaf map.
On the real line, take the sheaf of smooth functions. The germ at 0 of f(x) = x^2 and of g(x) = x^2 + e^{-1/x^2} (extended by 0 at the origin, which is smooth) are DIFFERENT germs even though all their derivatives agree at 0 — they disagree on every neighborhood of 0, so no shrinking can make them equal. The stalk at 0 is the ring of all such smooth germs.
Germs at a point: agreeing to infinite order is not enough; germs must agree on a whole neighborhood.
A sheaf map being iso on all stalks forces it to be an iso of sheaves — but this is special to sheaves; for mere presheaves equal stalks do not imply isomorphism. Stalks are the right notion of 'value at a point' only for sheaves.