Pets are machines that are irreproducible because they are highly customized over a long period of time with no record of how to exactly replicate the process. They must be managed individually. If a pet becomes damaged or corrupted, it must be carefully brought back into the desired state just as a doctor tends to a sick patient.
Cattle are machines that can be reproduced programmatically and are therefore disposable. If one of these cattle gets damaged or corrupted, it is wiped and rebuilt. To complete the analogy, when a single animal in a cattle drive is sick, it is killed so that the herd can keep moving.
Cattle-like systems make it easier to manage large numbers of machines. It is easier to mass-produce IT when machines are generic.
Desktops can be made cattle-like by starting them all the same via automation, and using directory services and other techniques to maintain their sameness. We can also reduce the number of vendors and models to make the repair processes more generic.
Servers have different challenges. The software each runs is usually very different. We can use containers and configuration management systems to automate the setup of these differences so that they can be reproduced by running the code again. More importantly, pet-like servers store irreproducible state: information that is not stored elsewhere (other than backups). We can design our services to separate out our state to specific machines so as to increase the number of cattle-like systems. State can be stored on a separate file server, database server, or external service.
We can also improve efficiency by making processes more cattle-like. A process should save any variations until the last possible moment. By keeping things generic at the start, we can mass-produce the start of the process.