Proof of Importance
The detailed overview can be found in NEM's Technical Reference document. In general, it looks like a variation of Proof of Stake that gives extra weight to accounts that are actively transacting with a lot of other accounts. The actual equations as to how the weight is calculated and how it affects who will be selected as a miner is detailed over several pages of the document.
The downside of putting extra weight on transactions between accounts on the network is that the system can be easily gamed, at least until we solve the "one-identity-per-person" problem. Creating accounts and sending transactions around is trivial, and figuring out the exact method of getting the maximum return to game the algorithm is only a matter of time. Even the document itself demonstrated that an attacker performing a sybil attack can boost their importance by about 10% in comparison to someone that just holds onto their tokens:
The sybil attack vulnerability of NEM reminds me of the old problem faced by Bitcoin mining pools back in 2011 - pool hopping. If a system allows any attacker to gain an advantage over honest players in the system, the algorithm needs to be changed.
As for the rest of the NEM system - I haven't yet had the time to explore it too deeply, but it also seems to suffer from the problem of being easy to front run by the miners (or harvesters as they appear to be called in NEM).