Can someone tell me what is meant by "layer 2 adjacency"?

Solution 1:

Layer 2 Adjacency, in Ethernet networks, refers to the idea that a packet send out on a segment can directly reach its destination without traveling through a device which would change the packet.

A simple example:
You've got two computers with a wire in-between; whatever comes out of one computer is essentially guaranteed to be receivable by the other computer. These devices have network adjacency.

More complicated:
You've got two computers with a L3 switch in between, both are assigned the same vLAN. Here again what one computer sends the other will receive without question.

Breaking adjacency
Two computers on the L3 switch again, but one is connected to a port assigned to a vLAN, the other is assigned to a trunk port. Now packets from the first computer will arrive to the second computer with a vLAN tag (assuming a typical setup here...)

Who cares? Well routing protocols do (they usually need to be able to figure out the network topology, and that discovery can be broken by non-adjacency), as well as a myriad of non-IP protocols. It's not all that common for these issues to come up on servers in a DC, but is certainly possible.