What is existence?

"A chair is still a chair, even when there's no one sittin' there" - Dionne Warwick

I’ll start by saying something about what existence isn’t: Existence isn’t a property for categorizing objects.

This is an idea I get from three philosophers: Gaunilo, Kant, and Quine. But it needs some explanation! 
Some things are horses; most things aren’t. We can sort the horses from the non-horses. 
Some things are dogs; most things aren’t. We can sort the dogs from the non-dogs.
Some things… well, I could go on, but you get the point.

Now, though, imagine trying to sort what exists from what doesn’t exist. That would be a daft task. There’s no task to perform, because everything exists!
That’s what I mean by saying that existence isn’t a property for categorizing objects.

Here’s another way to make this point
By definition, a unicorn is: a horse with a horn. So, if you want to know whether there are any unicorns, you need to try to find things which (1) are horses, and which (2) have horns. You don’t also need to double-check that (3) they exist!

And here’s that point about definitions, put in reverse. Let’s define a dracopug as: a fire-breathing pug which exists. Clearly, there are no fire-breathing bulldogs. So there are no dracopugs. It doesn’t matter that we tried to build existence into the definition of “dracopug”; our attempt adds nothing to the definition!

So, to repeat, existence isn’t a property that we can use to categorize objects. 
But the question I was meant to answer is: What is existence?
My suggestion—inspired by Frege—is that existence is a property of properties. But again, this needs explaining. 

Here are three ways to say the same thing: 

• dogs exist
• there are some dogs
• some things have the property being a dog

And here are three ways to say the same (different) thing:

• unicorns do not exist
• there are no unicorns
• nothing has the property being a unicorn

Now compare the third sentence on each of my two lists. Some properties (e.g. being a dog) have instances; some things (e.g. being a unicorn) don’t. And existence is the tool we use for categorizing properties along these lines.

With this in mind, I can answer your question! Existence is: the (higher-level) property which a (lower-level) property has, just in case there are things with that (lower-level) property.

Now, I’d be surprised if you found this answer completely satisfying. After all, I used the phrase “...there are things...” in my answer. I might equally well have said “...there exist things...”. So I’m using the notion of existence to explain existence!

I don’t think there’s any way to avoid going in a circle like this. Some concepts are just too basic—too fundamental—to be defined in other terms. Existence is one of those concepts.

My answer, then, wasn’t an attempt to define existence in a non-circular way. I really don’t think that can be done; so if you have no idea what existence is, I can’t help much! My answer is just trying to draw attention to the fact that existence isn’t a (lower-level) property of objects; it’s a (higher-level) property of properties.