[oakridge]

The article is a bit vague on these so I'll take a stab:

Random network are generated randomly, i.e. drawn from some distribution. An example would be the Erdos-Renyi (ER) network, where edges are drawn from a Bernoulli distribution between each node in the network. The simplest way to measure random networks is to determine the degree distribution. ER networks hence result to degree distributions that follow the Binomial distribution (Poisson as N->infty, Np->const). Since the edge probabilities are rather simple and drawn independently, one can take the ER network as sort of a "null model" of random networks.

Scale-free networks, on the other hand, do not follow this degree distribution. Rather, they follow a power-law distribution p(k) ~ x^-a. A process to explain this is that the network is generated by preferential attachment: nodes are more likely to be connected to nodes which are highly connected in the first place. Most of the discussion and the following arguments on this are in the article.