1) A dark pool. This is just another name for an exchange, but one that is not public. I don't know if NYSE runs one, but it wouldn't be surprising. The theory behind these is that they are provided as a service to large institutional investors to trade with each others outside the vagaries of the public markets for lots of reasons (mostly advertised as a way to not impact prices). The reality of these has been mixed at best. You have to opt into using a dark pool.
2) Payment for order flow. In this case your broker (not the exchange) has an agreement with a third party to sell your trades to them so that they can either trade with you directly or route the trade to an exchange. The reason they want to do this is that the aggregate of all the trades they are buying is not price opinionated (as opposed to hedge funds say) so the market maker can make the aggregate difference between all of them. This mechanism is largely how all discount brokers offer cheap or free trading, so it is likely a huge boon to you. A couple of things to note about this, your broker has to disclose it to you, I believe you are still required to get the NBBO price, & I believe you are required to be able to opt out by directing your order to be directly routed.
[edit] To clarify your question `jsprogrammer `tptacek is referencing #2 above. The reason the technical oddities & violations at NYSE don't have a huge impact on a retail investor is that unless otherwise stated, they are interacting with a wholesale market maker before their order hits any exchange, in a deal worked out with your broker and governed by legal obligations on both their parts.