What is a discrete set exactly? I'd appreciate the comments/answers including some examples in this regard. In addition, one may also ask that any infinite discrete set is a countable set?
 A: The two concepts have nothing to do with each other.
A metric space (more generally a topological space) is discrete if each point is isolated. For example, take the set of all real numbers (which, as you probably know, is uncountable) and define a new distance function
$$\operatorname d(x,y)=\begin{cases}
1\text{ if }x\ne y,\\
0\text{ if }x=y.
\end{cases}$$
This is an uncountable discrete space.
A space is countable if its points can be put in one-to-one correspondence with the natural numbers. For example, take the set of all rational numbers with the usual metric. It is countable but not discrete.
A: A topology has nothing to do with the set or the elements.  Any set whether it is the set $\{1,2,3 \}$ which is finite, or the set of real functions which is a higher uncountability than the reals, can be made into a topology that can be discrete or not.  A topology is how we define what subsets are open.  It has nothing to do with cardinality.
A discrete space is one in which the single point sets are defined to be open.
If, for $S=\{1,2,3 \}$ we declare: $\varnothing , \{1 \}, \{2 \},\{3 \},\{1,2 \},\{1,3 \},\{2,3 \},\{1,2,3\}$ to all be open sets then that is a discrete space.
If on the other hand we declare: $ \varnothing $ and $ \{1,2,3 \}$ are open but all the other subsets are not, then it is not a discrete space.
Even though they have the same elements.  The elements have nothing to do with which sets are open.
Likewise, if S=$\mathbb R $ and we declare the sets $\{x \}$ are all open then that is a discrete space. If we declare they are not open then that is not.
You might be saying, "Wait a minute... we can just declare whether a set is open or not?".  To which the answer is— Yes, we can... to a degree. (There are some restrictions.) That's what a topology is. It's a definition of openness combined with a set of elements. The actual set of elements has no say in what rules for openness we declare.
