In short: The statement "Property $P$ holds for a general hyperplane/line/etc" should mean that property $P$ holds for "almost every" hyperplane, line, etc. Now the cruicial point here is how to formalize the notion of almost every hyperplane - you have to model all hyperplanes: In other words, give them the structure of, say, a variety. Then you can say that some statement holds for almost every hyperplane if the property holds for a Zariski-dense open (Edit 05/08) subspace of the hyperplanes (in this model).
As Francesco already pointed out, this is exactly what the Grassmannian does for the linear subspaces of a vectorspace. It can be modelled as a projective variety via the Plücker embedding and then, it makes sense to say that some property holds for "almost every linear $n$-dimensional subspace".
Now this might help understand S. Carnahan's comment: The Hilbert Scheme is a construction that can serve as such a model for all closed subvarieties of a variety, hence allowing you to speak of a general hypersurface of degree $d$, for instance.