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On Wikipedia, the initial defection of a Cover in the Set-Theoretic Sense is given in $1.)$ as follows.

$1.)$ A cover of a set $X$ is a collection of sets whose Union contains X as a subset, the initial formulization of this is given in $1.2.)$

$1.2)$ $$C = \Big\{U_{a}: a \in A \Big\}$$ if $1.2$) is an indexed family of sets $U_{a}$, then $C$ is a cover of $X$ since in $1.2$) is an indexed family of sets following in $1.3$), we have the initial conclusion.

$1.3)$ $X \subseteq \bigcup_{a \in A}U_{a} $

My initial question from the definition given is I failed to understand 1-1.2) in simple terms, also what does an "indexed family of sets" mean ?

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    "indexed family of sets" may not be extremely relevant here; you may start with just a set of sets. Or start playing with the case that the index set $A$ is something simple, e.g., $A=\{1,2,3,4\}$. Then a covre is just as set $C=\{U_1,U_2,U_3,U_4\}$ with $X\subseteq U_1\cup U_2\cup U_3\cup U_4$. You might say that indexing the $U_i$ is just for notational convenience. With more understanding, the necessity (and usefulness) of such indexing may become more familiar and apparent – 2017-02-11

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I make no claim that this is the original inspiration for the use of the word "cover" in this context, but maybe it will help:

Imagine your set $X$ is a table top, and your sets $U_a$ are each pieces of paper on the table. Then the pieces of paper (i.e. the set $C = \{U_a \;|\; a\in A\}$) "cover" $X$ if you can't see the table top because the paper is blocking it all.