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Pets in the context of server management, refers to the traditional attitude of careful curation and nurturing of individual machines (as opposed to cattle).
Prior to the advent of extensive automation and configuration management, individual servers were set up by hand. A great deal of affection often developed for each individual machine. Often clever names were assigned, and failover procedures might be difficult or unrehearsed. The time cost of setting up a new server would be extensive. Carried to the extreme, a Pet could be a snowflake.
- EngineYard blog post which explores this idea in more detail.