Intuition for cofibration

The notion of a fibration has a nice geometric intuition of one topological space (a fiber) being parametrized by another topological space (the base) -- this is taken from the Wikipedia entry on Fibration.

Now, I would like to know if there is an analogous geometric picture for the situation of cofibrations. Like that of a parametrized inclusion of a space into another space (apologies if this is nonsense).

Yes, there are precisely the dual diagrams defining a cofibration. The way to think about them -- the way I think about them, at least -- is just that they're sufficiently nice inclusions, such that the subspace has a bit of "wiggle room". This is made precise in the definition of a "neighborhood deformation retract (NDR) pair", which you can read about in May's Concise Course in Algebraic Topology. Maybe the best example to keep in mind is when you turn an arbitrary map into a cofibration with the "mapping cylinder" construction; then it's obvious that the subspace has plenty of wiggle room.