Existence is treated one way in philosophy, but it is often spoken of in a variety of ways in other fields. The following article (at present) only speaks about the concept as it is treated by some philosophers.
'Exists', the word and concept, is often held to be indefinable in ontology--a branch of metaphysics and philosophy. This is not because 'to exist' is thought to be meaningless, but rather that an implication of existence is "built into" our ordinary language. For example, if I say, "A robin is visiting our bird feeder," it does not add any new information to add, "And by the way, the robin exists."
Frege and Russell, among many others, for similar reasons are well-known for their view that "exists" is not a (logical) predicate, or more precisely, not a first-order predicate--or, perhaps equivalently (and perhaps not), that existence is not a property. This has become the dominant but not the universal view in twentieth-century and contemporary Anglo-American philosophy. But, as G. E. Moore pointed out in an early essay, titled "Existence," it is a matter of some difficulty to say what exactly what this view amounts to.
The words (and concepts) 'existence' and 'being' are treated in slightly different ways in Western philosophy. Aristotle pointed out that there are various ways in which a thing can "be" and inaugurated ontology as a field with his notion that there are categories of being, such as substance, attribute, and acting-upon. Similar claims, however, are not as often made on behalf of existence. That is, sometimes when contemporary philosophers discuss the topic treat existence as a univocal, unambiguous concept, as if there were only one sense of 'existence'. This sense, if there is a single sense, is thought to apply (or not to apply) to physical objects, minds, God, numbers, possibilities, properties, and so forth--everything in which philosophers ask, "But does X really exist?"
The ambiguity of 'does not exist' is not often overlooked, however. That is, ontologists are fond of pointing out that there are various ways in which things can fail to exist: they can be fictional (Sherlock Holmes), imaginary (the golden mountain), legendary (Loch Ness Monster), mistakenly inferred (the ether), etc. The multiplicity of ways in which a thing can fail to exist has be so striking to some that it has been suggested that existence is, in fact, merely an "excluder" concept--used to classify items by what they are not (not fictional, not imaginary, not mistakenly inferred, etc.)--as 'real' is sometimes thought to be.
Though often not discussed under the heading of existence, disputes among realism, phenomenalism, physicalism, and various other metaphysical views concern what might be called the criteria for existence. For example, phenomenalism, generally speaking, is the view that everything that exists is mental. Most phenomenalists would want to deny that this claim is a definition of 'exists'; if phenomenalism were treated as a definition of 'exists', then others might accuse the view of trying to be "true by definition." Accordingly, it might be dismissed as a trivial exercise in redefining the ordinary concept of existence, which is, perhaps, of little interest to anyone. Exactly what relation, however, definitions (or analyses, or explications, etc.) and criteria have is an interesting and vexed question.