I've seen pet buttons mentioned around the forum but what are they? What's their use?

They're probably referring to custom buttons made for linking to pet profiles on your user profile.

You make a button graphic for your pet (or commission someone to make it for you) and upload it to an image host. Then you add a blank section to your user profile, and update it with code to display the button, linking it to your pet's profile.

so it replaces the pet headshots on your profile?

Pretty much, but it doesn't have to. The custom buttons are added to a "blank section", while the standard headshots are in a "pets" section. They can both be displayed at the same time, if you wanted to do that for some reason. Usually people remove the pets section if they're using custom buttons.

awesome, thank you!

That is pretty cool. How do you make them?