Something exists iff it is part of reality.
In one sense, this is a difficult term to define, since it somehow is true of
everything that is the case, and hence seems to suppose some sort of theory
about everything; in another sense it is easily done in terms of "reality", to
which one then shifts the problems.
There are some logical problems connected with the term, such as Kant's
question whether "existence" is a predicate, and the issue whether the
existential quantifier in logic, that is read as "there is", "there
exists" or "(for) some", and that may be defined in standard logic in terms of
the universal quantifier: "not every x is not P" =def "some x is P",
carries the full weight of all that one might mean by "exists"; and combining
both whether it makes sense to introduce, also in logic, statements to the
effect that "some things exist" - i.e. formally, writing "E!" for "exists": "(for some x)(E!x)".
One problem one has here is that it would follow that
"(for some x)(~E!x)" must somehow be true - of which one reading, namely with the particular quantifier as read in standard formal logic, is "there
exist things which don't exist". (See: Nothing.)
This again relates to problems raised by intuitionist philosophers of
mathematics, who believe that one should be
able to give a constructive proof of whatever one claims to exist, that
shows how one could produce it, at least in principle, that again relates to
statements like "there exist particular natural numbers that no one ever
thought of". For if you name such a one, you have thought of it.