The use of hedgehogs involves placing arrows in the dataset at various points, where the length and direction of the arrow directly corresponds to the length and direction of the vector at that point (interpolating if neccessary). Further, the arrows can be coloured, according to some other variable, most commonly the magnitude of the vector at that point.
Hedgehogs have the advantage that they are very easy to construct, both in terms of human effort and computational complexity. It is very easy to construct a simple network with which to visualise data using hedgehogs and the visualisation should proceed quite quickly. This allows good interaction and probing of the data and is excellent for obtaining a first look at vector data. Further, arrows are the canonical way of representing vector data, which makes them comfortable to work with.
There are disadvantages, however. The hedgehogs can be quite visually complex, which means that usually only small subsection of the data may be hedgehogged at any one time (for example, a single slice as opposed an entire data cube). Further, the use of colour to represent magnitude, while very effective, can be an additional redundancy. Hedgehogs are also quite poor at showing the vorticity (twist) of a vector field.