Do you have to refrigerate butter?

There are a couple of things going on with butter that makes it safe to leave it unrefrigerated. Not only is basically fat and water, it's highly saturated fat. The level of saturation means it's more difficult for it to oxidize (turn rancid) than unsaturated fats.

The other factor is salt, which is often added as a preservative. Salted butter has a shelf life almost double that of unsalted butter.

While butter does not need to be refrigerated, the colder it's stored, the longer it will last. Protect it from light and air and pay attention to when it begins to turn rancid. Rancid fats are harmful to one's health.

Is there something about butter that makes it safe to leave out. It is dairy after all.

I always keep a stick of butter at room temp in a covered dish. I just smell it before I use it.

I have used the Emile Henry butter pot for 5 years and I love it. The butter stays on the counter, soft and spreadable, and I never have any problems with it going bad. However, this method of storage will not work in a hot and humid environment. I live in Colorado and use my butter pot year-round.
http://www.emilehenryusa.com/Butter-Pot-Azur-plu538621.html

I second SeaJambon. In East Tennessee, where summer and even spring temps are very warm and the humidity is almost always high, I've tried using a butter bell, but my butter gets rancid pretty quickly. I would imagine that in more temperate climates you might not have such an issue.

Kitchen/home heat is the issue. In the Pacific NW (where I currently live and grew up) we routinely keep butter in an uncovered dish on the counter, perfect for easy spreading on toast. When I've lived in other areas (like DC and Texas) I discovered that approach encouraged either melted or rapidly rancid butter. Bottom line: heat and humidity in your location needs to be taken into account. I haven't used any of the "butter keepers" (like the bell) mentioned above -- frankly because there is no need here (other than perhaps protecting the butter from the cat, but that's another issue!) -- but have heard great things about them from friends who live in other areas.

There is a device called a "butter bell" that can help to some degree. It's a stoneware piece. You put cold water in the bottom and replace the top. I've had middling results with it---but at least it looks nice on the table.

How hot is your kitchen? Maybe not in extreme summer without a/c....

No -- but I'd keep it covered, and use it within a few days.