First off, those comments (and the book in general) are actually about toposes, not general categories, and should be interpreted in that light. (This is something I've only come to realise recently, despite having read the book a couple of years ago.) Therefore one should look to categories of sheaves to make sense of the remarks.
And I am also puzzled by what exactly they meant by T=1. Did they mean "T= the category's terminal object"? Or "T= the category's separator object?" Or "T= a/the singleton object"? In Sets all these descriptions apply to 1, but it is not clear to me which of them is the basis for the authors' comments.
$1$ always means the terminal object in this book, if I remember correctly.
The notion of ‘singleton’ is very set-centric. It is possible to generalise this idea to toposes, but this is not what is meant here. Toposes in general do not have a separator object either; those that are separated by $1$ (and with $1$ not isomorphic to $0$) are called well-pointed.
Furthermore, it means that the authors are either (1) implying that the term injective can be meaningfully applied only to morphisms in categories that have a "1" object (with 1 = terminal/separator/singleton, depending on what the authors meant), or else (2) implying that every concrete category has one such object.
As already mentioned, not every category has a separator. Nor does every category have a terminal object. One can study the implied definition in any category with a terminal object, but it is not necessarily useful. For example, in categories where the terminal object is isomorphic to the initial object, e.g. $\mathbf{Grp}$ or $\mathbf{Ab}$, there will always be exactly one arrow $1 \to X$ for any object $X$. It is most profitable when the category is a topos. In the traditional case of categories of sheaves, an arrow $1 \to X$ is called a global section, so the comment is simply saying (for this case) that ‘a sheaf may not have enough global sections’, which is not surprising at all.
Indeed, consider the category $\mathbf{Set}^2$ consisting of pairs of sets and pairs of maps. The terminal object in this category is a pair of singleton sets. Consider the pair $(\emptyset, \{ * \})$. There is no arrow $(\{ * \}, \{ * \}) \to (\emptyset, \{ * \})$ because there is no map $\{ * \} \to \emptyset$. But clearly $(\emptyset, \{ * \})$ is not ‘empty’ (both in the intuitive sense and in the sense of not being an initial object). The book might call this an example of ‘not having enough global elements’. The way to rectify this is to consider ‘generalised elements’, which in the language of sheaves is akin to the notion of a local section. But notice that this category has a separating set, namely the pair of pairs $(\emptyset, \{ * \})$ and $(\{ * \}, \emptyset)$. It is tempting to take the coproduct of these two to try to make a single separating object, but we have already shown that doesn't work!
Now, to answer your first question. Note that monicity can be defined as a limit: an arrow $f : A \to B$ is monic if and only if the pullback of $A \xrightarrow{f} B \xleftarrow{f} A$ is the identity map. I will discuss only concrete categories here and ‘injective’ is reserved for maps of sets.
Necessary conditions. The above characterisation is useful because we can now understand preserving monics in terms of preserving limits. Thus, if the underlying set functor of a category preserves finite limits (or even just pullbacks), the underlying map of a monic arrow must be injective. This happens, for example, when the underlying set functor has a left adjoint (the ‘free object functor’). This means, for categories of where is a reasonable notion of ‘free object generated by one element’, the underlying map of a monomorphism must be injective.
Sufficient conditions. Similarly, if the underlying set functor reflects limits, then every arrow which has an injective underlying map must already be monic. This is typical in categories of algebraic objects and is true in particular for $\mathbf{Grp}$, $\mathbf{Ring}$, $\mathbf{Vect}$. Note, however, that the underlying set functor for $\mathbf{Top}$ does not reflect (or create) limits, so one has to search for other ‘reasons’ why a monic arrow in this category is the same thing as an injective continuous map. In fact, when the underlying set functor is faithful (i.e. injective on arrows), if the underlying map of an arrow is injective, then the arrow must be monic. The easiest way to see this is to use the fact that the functor $\mathrm{Hom}(C, -) : \mathbf{C} \to \mathbf{Set}$ preserves monics. (Exercise!)
It is obvious that injective (respectively, surjective) group homomorphisms are monomorphisms (respectively, epimorphisms).
Monomorphisms in the category of groups are injective maps. Indeed, suppose $\phi\colon G\to H$ is a monomorphism; consider $\alpha\colon\ker\phi\to G$, the canonical injection, and $\beta\colon\ker\phi\to G$, $\beta(x)=1$. Then $\phi\circ\alpha=\phi\circ\beta$: what does $\alpha=\beta$ entail?
Epimorphisms in the category of groups are surjective, but this is a bit more difficult to show (one needs to define an action on the set of cosets of $H$ by the image of $\phi$).
The standard example of a nonsurjective epimorphism in a category is the embedding $\mathbb{Z}\to\mathbb{Q}$ in the category of rings, which is both a monomorphism (obvious) and an epimorphism (try it).
Best Answer
The prefix "epi-" in Greek has several meanings, but a common one is "upon, over". This is similar to the meaning of the prefix "sur-" in French, which was the origin of the term "surjective", introduced by Bourbaki. As such, both give the meaning that the function/morphism "covers" all of its range.