Spatial relationship where two features share a common border under some interpretation - ie, are 'next to' each other.
Adjacency refers to a spatial relationship where two features share, to some interpretation, a common border. It can be interpreted in various ways - perhaps they only share a single point, or there may be a small gap between.
Consider a regular vector grid. Adjacency can be used to determine which cells surround a given cell. In some cases the cells sharing only a corner might not be considered adjacent, because they do not share a full side.
Consider a group of lots in a block in an urban area. Adjacency indicates which lots or buildings are next to which others. If there is a gap between lots, such as for an alley, they may or may not be considered adjacent depending on criteria and context.