I am reading algebraic topology on my own. I am facing some problem in notation.
Consider one of the covering space of circle, that is $\mathbb R$. And consider the covering map $p: \mathbb R \rightarrow S^1$, which is defined by $x \mapsto [x \text{mod 1}]$.
What does one mean by $[x \text{mod 1}]$ in this context?
There are several different but equivalent definitions of the unit circle. Arguably the most immediate one is the one as subspace $\mathbb S^1 := \{x\in \mathbb R^2 \mid \Vert x\Vert = 1\}$. It is common to identify the real plane with the complex numbers, so this becomes $\mathbb S^1 := \{z \in \mathbb C\mid \vert z \vert = 1\}$. This has the benefit, that we can write down things in terms of complex numbers instead of having to deal with $x$- and $y$-coordinates.
Another way to define the circle is by saying that it is a line segment, whose endpoints get glued together. Mathematically this boils down to defining the quotient space $\mathbb S^1 := [0,1]/\{0,1\}$. While this is a very good definition in terms of intuition, from a technical standpoint it turns out that it would be nice to have a quotient by a group action. We can accomplish that by considering the circle as $\mathbb S^1 := \Bbb R/\Bbb Z$, where $\Bbb Z$ acts on $\Bbb R$ by translation, meaning $n \bullet x = n+x$. Note that we could also fix some real number $c>0$ and consider the quotient $\Bbb S^1 := \Bbb R/c\Bbb Z$, where $c\Bbb Z = \{cn \mid n \in \Bbb Z\}$ also acts by translation $cn \bullet x = cn+x$. Up to homeomorphism this would not make a difference, we are just used to $c = 1$.
Let me briefly dive into what the last definition means in detail. The points in the space $\Bbb R/c\Bbb Z$ are equivalence classes of elements in $\Bbb R$, ie. sets of the form $[x] = \{y \in \Bbb R \mid \exists n \in \Bbb Z: y = nc+x\}$. There are many different ways to denote these equivalence classes. In the given situation $[x]_{c\Bbb Z}$, $[x \mod c]$ or $x + c\Bbb Z$ would be common notations for what I denoted $[x]$ above.
It is a nice exercise in point-set-topology to prove that all of the different definitions of $\Bbb S^1$ I gave are actually equivalent. It seems to me that the source you cite has the identification $\Bbb S^1 = \Bbb R/\Bbb Z$ either as a result or even as definition of the circle.