Category with only one object. Suppose we have the category with only one object - the group G. 
Why can we think of the morphisms in this category as of the elements of the group G?
I would be very grateful for explanation.
 A: In general the morphisms of a category with one object form a monoid. The existence of inverses isn't guaradnteed. But any group can be viewed as a category with one object (again, actually any monoid can). Let us call the object $A$, it doesn't really matter. By definition of a category the morphism set $\text{Hom}(A,A)$ has a binary operation given by composition. Again by assumptions of a category, this composition is associative, and there is a two-sided identity. So it is a monoid. In particular if we have some monoid $M$ in mind we may take the category with $\text{Hom}(A,A) = M$ and where the composition of two morphisms is simply their product in the monoid. 
A group has inverses, so a group will give you a category in which all the morphisms are invertible. 
A: As pointed out by JHance, the endomorphisms of the unique object of a one-object-category do in general only form a monoid.
If you start with a group $G$, you can consider the category with one element that has as endomorphisms the elements of $G$ and composition of morphisms in this category is given by multiplication in the group $G$.
A natural generalization of this situation to categories with more than one element are groupoids. A groupoid is a category in which every morphism is an isomorphism. If there is only one object in the category, this is nothing but a group.
