In maths, the notion of germ is used to definition “generic” of functions, in other hand, germ is used to say about mostly functions on topological spaces.
Consider to a point, we need to get information from that point, so we take a neighborhood of that point, assume that the point is .
With a smooth function on manifold , germ of smooth functions near is a pair such that is smooth on local neighborhood of .
Two pairs and is equivalent if there is an open subset such that . So a germ of smooth function near is an equivalent class of that pairs.
(to be cont.)
References: Wikipedia, planetmath.org