What things can be defined in terms of universal properties? We can define some mathematical objects using universal properties, for example the tensor product, the free group over a set or the Stone–Čech compactification.
I'm wondering about how to develop my intuition so that I can spot a thing I can define using a universal property when I see it. 
It seems clear that a necessary condition on the object is that it is unique. For example for two topological spaces and a function $f: X \to Y$ we cannot define continuity of $f$ in terms of universal properties since there are many functions $X \to Y$ that are continuous. But is it sufficient for an object to be unique (up to unique isomorphism) in order for it to be definable using universal properties?
To summarise into one question: what characterises objects that can be defined using universal properties?
Thank you.
 A: Everything can be defined in terms of universal properties! Namely, every object is the unique object, up to unique isomorphism, which is exactly like that object. More precisely, let $x$ be an object in a category. Then $x$ is the universal object equipped with an isomorphism to $x$ (namely the identity). (As Martin Brandenburg says, by the Yoneda lemma this is equivalent to saying that $x$ is the universal object such that the functor it represents is equipped with a natural isomorphism to $\text{Hom}(x, -)$, so this is not as vacuous a statement as it first appears if you first describe $\text{Hom}(x, -)$ some other way and then ask whether it is representable.) 
More seriously, if you want to develop your intuition, I think the best thing is to become familiar with a lot of examples. Look at your favorite categories and try to figure out what the products, coproducts, limits, colimits, etc. are in that category. Look at your favorite functors and try to figure out if they have left or right adjoints (equivalently if they are left or right adjoints). If they map to $\text{Set}$, figure out if they're representable and, if so, what the representing object is. And so forth. Gradually you'll become better able to recognize when something is or ought to be defined universally. 
An important example to keep in mind is the tensor product $V \otimes W$ of two vector spaces. This has a universal property, but it's not obvious: the tensor product is neither the product nor the coproduct in $\text{Vect}$. (The actual universal property comes from a formalization of what it means for a map to be bilinear; one way to do this comes from the tensor-hom adjunction.) 
If that's old hat for you, try the tensor product $A \otimes B$ of two noncommutative rings. This also has a universal property and it's also not obvious: the tensor product is neither the product nor the coproduct in $\text{Ring}$. (But there's also no tensor-hom adjunction here!)
A: A universal property of an object $X$ in a category is just an isomorphism between $\hom(X,-)$ (or $\hom(-,X)$) and another interesting and more concrete functor. Since "more concrete" is not really precise, this essentially says that every object $X$ has a universal property, but a very boring one.
So the question is really: When do interesting universal properties occur? Well it is the other way round: Given a functor $F$, which often encodes some classification or comparison data, one may ask if it is representable. The Yoneda Lemma tells us that every representation of a functor is unique up to isomorphism. But the functor $F$ is more fundamental than the representing object itsself (after all, it doesn't have to exist!).
For example, when you do linear algebra, you naturally stumble upon bilinear maps $M \times N \to T$. You would like to classify them via homomorphisms. That is, you ask if the functor of bilinear maps $M \times N \to (-)$ is representable. And indeed, this is true, and the representing object is the tensor product $M \otimes N$. But many properties of the tensor product are just consequences of trivial properties of the functor of bilinear maps (for example $M \otimes N \cong N \otimes M$ just says that bilinear maps on $M \times N$ correspond to bilinear maps on $N \times M$ via a switch).
Of course, universal properties are abundant in pure mathematics. It would be nonsense to try to summarize them here or to attempt any rule how they appear. You should learn the examples first.
