This article is from the Nonlinear Science FAQ, by James D. Meiss firstname.lastname@example.org with numerous contributions by others.
(Thanks to Hawley Rising for contributing to this answer)
Generic in dynamical systems is intended to convey "usual" or, more properly,
"observable". Roughly speaking, a property is generic over a class if any
system in the class can be modified ever so slightly (perturbed), into one
with that property.
The formal definition is done in the language of topology: Consider the class
to be a space of systems, and suppose it has a topology (some notion of a
neighborhood, or an open set). A subset of this space is dense if its closure
(the subset plus the limits of all sequences in the subset) is the whole
space. It is open and dense if it is also an open set (union of
neighborhoods). A set is countable if it can be put into 1-1 correspondence
with the counting numbers. A countable intersection of open dense sets is the
intersection of a countable number of open dense sets. If all such
intersections in a space are also dense, then the space is called a Baire
space, which basically means it is big enough. If we have such a Baire space
of dynamical systems, and there is a property which is true on a countable
intersection of open dense sets, then that property is generic.
If all this sounds too complicated, think of it as a precise way of defining a
set which is near every system in the collection (dense), which isn't too big
(need not have any "regions" where the property is true for every system).
Generic is much weaker than "almost everywhere" (occurs with probability 1),
in fact, it is possible to have generic properties which occur with
probability zero. But it is as strong a property as one can define
topologically, without having to have a property hold true in a region, or
talking about measure (probability), which isn't a topological property (a
property preserved by a continuous function).