I was looking at a Meta post Fizz posted, and he had a discussion with another Databse user about these two services and whether people are allowed to discuss it in the Database, and Fizz said they are. But what exactly are these two things?
Wiimmfi is a whole new online server that replaces the now defunct Nintendo Wi-Fi Connection, and revives online play for almost all Nintendo DS and Wii games that had online capabilities! Initially designed for Mario Kart Wii, it expanded to cover almost all games up to this day!
Poké Classic Network provides GTS, battle videos, and other related services for Generation IV and V games. It runs in combination with Wiimmfi and Kaeru WFC which provide general purpose NWFC emulation and SSL offloading, respectively.
So basically both are servers that make playing online functions on (Pokémon) games on a DS and the wii possible.
Source, sauce.