A germ is an element of (a total space of) an etale space or equivalently an element in some stalk of a sheaf (all stalks together form the total space of the etale space of the sheaf). Exactly what this means depends on which sheaf one is considering.
This general description of what a germ of some sheaf at some point is can be extracted from stalk, although that article is pretty abstract right now.
More generally, the notion of stalk makes sense in any topos that need not be a Grothendieck topos of sheaves by way of the notion of point of a topos. Generally a germ is an element in the stalk of an object of a topos over some point of the topos.
Originally, the term came from geometry, where sheaves of (continuous, smooth, holomorphic etc.) functions or more generally, sections of a (say fibre) bundle $\xi:E\to B$ were considered; germs in geometrical intuition are always germs of something (of a section of the sheaf in some neighborhood of a point, but also elements of a colimit at a point of local sections of an original presheaf which is not always a sheaf; though the germs a posteriori make a sheaf, they can be considered or defined in relation to an original presheaf which is not necessarily a sheaf). Germs of sections are defined as the elements of the colimit sets of the appropriate sets of sections $\Gamma_U \xi$ where the colimit is over all open sets containing $x\in B$ with inverse inclusion (inverse because a presheaf of sections of a bundle is a contravariant functor).
In other words, germs of sections at a point $x\in B$ are equivalence classes $[U,s]$ of pairs of the form $(U,s)$, where $U\ni x$ is an open set in the base $B$ and $s:U\to E$ is a section defined over $U$; two sections $(U,s)$, $(U',s')$ in $colim_{U\ni x} \Gamma_U \xi$ are equivalent if there is a smaller $W\subset U\cap U'$ and $s|_W=s'|_W$. This construction of germs of sections of a bundle over $B$ (object of the slice category $Top/B$) leads to an adjoint pair of functors between $Top/B$ and the category of presheaves of sets over $B$ which restricts to the equivalence of the category of etale spaces $Et_B$ and the category of sheaves over $B$.
For example, take the sheaf of continuous (say, real-valued) functions on some space $X$. Then every partial function $f$ defined on a neighbourhood of any given point $a$ in $X$ defines a germ at $a$. Furthermore, the germ of $f$ equals the germ of $g$ if and only if $f = g$ on some neighbourhood $U$ of $a$; note that $U$ must be contained in the intersection of the domains of $f$ and $g$, but it may be smaller yet.
For a modern example of this kind, with only $1$ stalk, consider the nonarchimedean field of germs of holomorphic functions at the origin of the field $\mathbb{C}$ of complex numbers, which plays an important role in mirror symmetry as the base field for the geometry of families of Calabi-Yau manifolds in the large volume limit (cf. Kontsevich, Soibelman doi:10.1007/0-8176-4467-9, arxiv:math/0406564v1).
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
smooth functions | derivative | Taylor series | germ | smooth function | ||||
curve (path) | tangent vector | jet | germ of curve | curve | ||||
smooth space | infinitesimal neighbourhood | formal neighbourhood | germ of a space | open neighbourhood | ||||
function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||
arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||
Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||
symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |
R. Godement, Topologie algebrique et theorie de faisceaux, Paris 1958
Masaki Kashiwara, Pierre Schapira, Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006)
Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic, Springer 1992