The trivial example of such a property is finiteness: if $A$ is any infinite set, then all the finite subsets of $A$ are finite but $A$ itself isn't finite.
A slightly more interesting example is boundedness (in the context of metric spaces): every finite subset of a metric space is bounded, but there are metric spaces of unbounded diameter.
However, to appreciate the compactness theorem I believe it's most helpful to think about an example coming from logic - but not first-order logic! In second-order logic (working in the language of arithmetic) we can write the sentence "Every nonempty set has a least element;" when we add a couple more axioms of basic arithmetic, we get second-order Peano arithmetic, which is categorical in the sense that it has a unique (up to isomorphism) model, namely $\mathbb{N}$.
But this means compactness fails for second-order logic! Consider the language consisting of the usual language of arithmetic together with a new constant symbol $c$, and let $T$ be the theory in this language consisting of second-order Peano arithmetic together with, for each $k\in\mathbb{N}$, the sentence $$c\not=1+1+...+1\quad\mbox{($k$ times)}.$$ Then $T$ cannot have a model, even though every finite subset of $T$ does have a model (this is a good exercise).
