Let me try to give some info. Hope I didn't ramble too much and that is followable, this article exposes it better than I do:
http://news.sciencemag.org/math/2015/11/mathematician-claims-breakthrough-complexity-theory .
The graph isomorphism problem, which is that given any two finite graphs you can say whether they are the "same" (up to a relabelling of the edges), that is called isomorphic.
Technically this means there is a function between the two that is one to one (bijective) and preserves edges, i.e. two vertices have an edge between them in one graph if and only if they do in the other one.
They may look different but they are essentially equal, and share all properties. Eg the two below are isomorphic:
This problem is one of a few that is not known to be solvable in polynomial time nor NP-complete, which is almost certainly the most important open problem in complexity. P type problems are considered "easy", whereas NP-complete are at least as hard as any other NP problem, thus some of the hardest out there.
And now there is this algorithm by Babai which reduces it to quasi polynomial time, that is a much quicker than previous algorithms in the sense that it is almost polynomial in time, thus brings the complexity of the problem much closer to polynomial time (exp(polylog(n))). As such, it places this problem in an area of the problem much closer to P, whereas it was thought it was NP.
For applications, the size of networks and computational speed is probably to the point that this is not too big of an improvement, as the "slower" algorithms still tend to get it done quickly enough. Besides, knowing whether two networks are the same is not usually the question, although some others reduce to this one. But often it is only after the result comes in that new ways to use it are thought of (often coming from the ideas used in the proof of the statement itself).
