Function of a component in a Ethernet Board
It's a transformer (really several transformers and common mode chokes in single package). Often these days it is contained within the jack, but it's probably a bit cheaper to have it external. 

Here are some options available for 10/100 Ethernet: 



One of the main functions is electrical isolation of the differential signal wires in the Ethernet cable from the computer board. Both sides are transformer-isolated so the communication link is quite immune to EMI and it causes little EMI because the currents are balanced between the halves of the twisted pair. 

The isolation is demanded by the relevant Ethernet standard. It's actually possible to bypass the transformers and capacitively couple a (short) communication link, but fortunately the wise ones have required all links have transformer coupling, so the cost is very reasonable (it started out at more than $1,000 USD per node, but volume and Asian production worked its magic) and we have rather reliable communications.