a node that can be reached in one step by traveling an edge

When processing a category hierarchy in a strict tree manner, a node's neighbors are its previous and next nodes, according to the logic of a pre-order tree traversal (this is a concept from mathematical tree theory). For example, if a category hierarchy were displayed as an infinitely deep table of contents, then a node's neighbors would be those listed immediately above (previous) and below (next) it.

See Peer

Adjacent system reachable by traversing a single subnetwork. An immediately adjacent router. Also called a peer.

Another router with which implicit or explicit communication is established by a routing protocol. Neighbors are usually on a shared network, but not always. This term is mostly used in OSPF and EGP. Usually synonymous with peer.