Depending on the country you fly to and the airline, you can take your pet as an additional piece of luggage. This is treated the same way as if you were taking a pair of skies, surfboard or other large items. This is by far the cheapest option for pets to travel with big cost savings, however not all countries accept this type of entry (UK, Hong Kong, Australia, New Zealand, UAE to name the popular destinations).
Just like any other baggage, if you don’t fly with it then the airline will not take. In this case, if you are travelling to one of those countries listed above or are unable to be on the same flight then we would have to send as manifest cargo (like sending a package with a courier service). Pets would travel in the same part of the aircraft but they would be processed via the cargo terminals and not the passenger terminals. Sending pets as manifest cargo has always been the more costly way of sending your pets and especially now during the covid pandemic.
We also made a chart on this here.