Singletons (sets with single element): what's special about them? I am asking a broad question. I don't need precise definition. I just want to get "big picture" of why singletons considered like something important, especially when it comes down to category theory? 
Why, for example, 2-elements sets are not that special? 

Citing the comments as an answer:


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*In category theory (especially the categories of sets, and of topological / metric spaces), the defining property of singleton sets is that they are terminal objects:


There is exactly one function from any set to a one-element set.


*Secondly note that the elements of set $X$ correspond with arrows $\ast\to X$ where $\ast$ stands for a singleton. So $X$ corresponds with the homset $\mathcal Set(\ast,X)$. That gives possibilities to "catch" the theory of sets in the theory of categories. In other words


For any set $X$, its elements are in a natural one-to-one correspondence with functions $\ast\to X$ where $\ast$ is a one element set.


*2-element sets are also special in the category of sets, as these can be used as subobject classifier:


For any set $X$, its subsets are in a natural one-to-one correspondence with functions $X\to T$ where $T$ is a two element set.

