The Two Sense Theory of Exists
First level predicates are said of individuals; i.e., “Bob exists”.
Second level predicates are said of kinds; i.e. “Men exist”.
Another way to think about the difference is this. Objects fall under first level concept whereas a concept falls under second level concept. Second level predication (by extension second level properties) are thus properties of concepts.
Some philosophers claim existence is not meaningfully attributed to individuals but only to concepts, and because of that all talk of existence can be replaced with the existential quantifier: essentially, to say something exists is just to say there is at least one of something; something that falls under the concept rabbit, for example.
However, existence is meaningfully attributed to individuals — for example, of Thumper the rabbit, hopping about in my backyard — hence not all talk of existence can be replaced with the existential quantifier.
A better view is the two-sense theory of existence. Philosophers maintaining the two-sense theory of existence bring forth the distinction between exists in the “actuality” sense (attributed to individuals) and exists in the “there is” sense (attributed to concepts or kinds).