Brendan,
The ESP8266 is a microcontroller with built in wifi and a bunch of other stuff that I don’t even pretend to understand (Espressif product page). It can be procured as a bare module, or as a full dev board. It can be programmed via the Arduino IDE and acts a lot like an Uno in practice.
Arduino Uno sketches with minor tweaking
Craig did a quick tutorial here on how to get the code correct. If I can do it, anyone can.
Craig (@kreggly) just posted that he believes you should be able to get the ENC28J60 to work here. So maybe all this discussion is for naught.
Hope this helps,
-Ian