Butter should be kept refrigerated as keeping butter refrigerated helps keep the butter fresh longer and extends the shelf life of the butter.
Salted butters can safely sit out at room temperature in a covered dish for a few days, which can make the butter easier to spread.
Unsalted butters spoil or go bad faster and require refrigeration.
Whipped butter should also stay chilled and refrigerated to prevent spoilage.
You can leave even unsalted butter out at room temperature for a few hours if you are using the butter within that time.
Although if not using the unsalted butter within a few hours it should be refrigerated.
Other than leaving the unsalted butter out for a few hours, the unsalted butter should always be kept refrigerated as it lacks the preservative qualities of the salt.
Butter basically stays fresher for longer when the butter is kept cold.
For best quality of the butter it should be kept refrigerated, but leaving a small amount of butter out and covered for short periods is okay.
I usually keep a stick of butter out on the counter covered or a tub of butter out on the counter covered if using it within a few hours or within a few days and it has been okay.