What is the definition of "cattle not pets"?
Randy Bias chronicles the history of the term stating that it probably originated in 2011 or 2012 when Bill Baker used the analogy when describing "scale-up" vs. "scale-out" architectural strategies. Bias adopted this into his presentations about cloud architectural patterns:


  In the old way of doing things, we treat our servers like pets, for example Bob the mail server. If Bob goes down, it’s all hands on deck. The CEO can’t get his email and it’s the end of the world. In the new way, servers are numbered, like cattle in a herd. For example, www001 to www100. When one server goes down, it’s taken out back, shot, and replaced on the line.


Bias continues to define Pets as


  Servers or server pairs that are treated as indispensable or unique systems that can never be down. Typically they are manually built, managed, and “hand fed”. Examples include mainframes, solitary servers, HA loadbalancers/firewalls (active/active or active/passive), database systems designed as master/slave (active/passive), and so on.


and cattle as


  Arrays of more than two servers, that are built using automated tools, and are designed for failure, where no one, two, or even three servers are irreplaceable. Typically, during failure events no human intervention is required as the array exhibits attributes of “routing around failures” by restarting failed servers or replicating data through strategies like triple replication or erasure coding. Examples include web server arrays, multi-master datastores such as Cassandra clusters, multiple racks of gear put together in clusters, and just about anything that is load-balanced and multi-master.


Fundamentally, what Bias and Baker are trying to convey is there has to be a transition from how we treat servers from being "Unique Snowflakes" with names and emotional attachments, to a model whereby if we have a problem with the server we create a replacement and destroy the problematic server.

Finally, it is probably worth mentioning that in regulated environments taking a server out the back and shooting it may not be optimal.  In these cases it is often advantageous to "freeze" the server, for example using docker pause to freeze a container.  This can then be used to perform a Root Cause Analysis as part of the Incident or Problem Management Process.