The configuration model is a method for generating random graphs that match a specified degree sequence — the list of how many connections each node has. Unlike the Erdős-Rényi model, which produces a Poisson degree distribution, the configuration model can reproduce any degree pattern, including the heavy-tailed distributions of scale-free networks. It works by assigning each node a number of "stubs" equal to its desired degree and then pairing stubs uniformly at random to form edges.
The model is the workhorse of network null-model construction. By comparing a real network to a configuration-model randomization, researchers can determine which properties — clustering, assortativity, community structure — are consequences of the degree sequence alone and which require additional mechanisms. The configuration model reveals that many apparently complex network properties are mathematically redundant: they are implied by the degree distribution and contain no additional information about generative process.
A subtlety: the simple configuration model allows multi-edges and self-loops. Rewiring algorithms that preserve degree sequences while forbidding these artifacts are often preferred in practice, though they sacrifice the analytic tractability of the stub-matching approach.
The configuration model is network science's most important diagnostic tool not because it tells you what a network is, but because it tells you what you already knew. Any property that survives configuration-model randomization is a genuine signature of structure; any property that vanishes was an illusion cast by the degree sequence. Most network papers would be better if they started here instead of ending here.