What is $\mathbb{R}/\mathbb{Z}$ going to be? Still kind of kind of not getting the hang of this.
When we see $\mathbb{Z}/\mathbb{2Z}$ for instance, I can relate it to the set of integers modulo $2$.
I look at $\mathbb{R}/\mathbb{Z}$ and I am not sure. But consideinrg the notaton analogous to $\mathbb{Z}/\mathbb{2Z}$ I think

$\mathbb{Z}/\mathbb{2Z}=\{0,1\}$. This can be thought as "the integers that are in $\mathbb{2Z}$" and "those that are not." So $\mathbb{R}/\mathbb{Z}$ means to separate "the reals that are in $\mathbb{Z}$(i.e. te integers)" and "those that are non0integers". 

So, is it $\mathbb{R}/\mathbb{Z}=\{[x],[y]:[x] \in \mathbb{Z}, [y] \in \mathbb{R}\setminus\mathbb{Z}\}$ so just as in $\mathbb{Z}/\mathbb{2Z}$, only two distinct elements? integers and non-integers??

Update: I have been commented that there is implicitly defined some equivalence relation $x \sim y \Leftrightarrow x-y \in \mathbb{Z}$. In more general terms, I would still be confused, for instance, when we define a quotient of sets to define a cone on a set $X$, apparently it is denoted
$$X \times I/(X \times \{0\})$$
here, I understand it to be all $(x,t)$ where $t=0$ are considered "equal" i.e. the equivalnce relation is that $(x,t) \sim (y,s) \Leftrightarrow s=t=0$. In this case it seems, in normal human langauge, it can translated to

"we will define an equivalence relation on $X\times I$, which we symbolize by '$/$' and the specific relation is that, if the pairs $(x,t),(y,s)$ have $0$ on their second entry, we treat them the same"

In other words, any $(x,t) \in X \times \{0\}$ are all the "same" here. Unlike the case in $\mathbb{R}/\mathbb{Z}$ which we define in the equivalence relation an implicit "-" operation $x-y$ going on, the above quotient is rather straightforward. I need not think $(x,t) \sim (y,s) \Leftrightarrow t-s=0 \text{ or } s-t=0$ or somewhat in a similar way.
Namely, the cone is about "whether it's in the set $X \times \{0\}$ or not" whereas $\mathbb{Z}/\mathbb{nZ}$ isn't. It has some additional operation "-" defined on it.
And to add some more...If $N$ is a normal subgroup of $G$, then we have  $G/N=\{\text{left or right cosets of }N\}$, which I can barely relate to the discussion we've had here. Sometimes there are some operations "$+-\times /$" defined for an equivalence relation, sometimes there isn't, sometimes it's whether or not it's in the set given, sometimes it's not, sometimes it requires an explicit definition such as for $G/N$.
I mean, in general, I have sets $X,Y$. I say what is $X/Y$? Is there a way to rigorously define this? Seems like I can say "well, maybe all elements of $X$ that are in $Y$ can be seen as equivalent?" or "any $x \times y \in Y \Leftrightarrow x \sim y$?" or "any $x-y \in Y \Leftrightarrow x \sim y$?" What are the rules?
But they use the same notation "/" and further, we are in the world of set theory, basically. In different areas of math I understand the same notation meaning different things sometimes, but this shouldn't be the case, should it? So I don't see how I can distinguish the "instruction" of one on another, I simply see no pattern in defining the quotient set/space....
 A: A concrete realization of $\mathbb{R}/\mathbb{Z}$ is given by the map $t \mapsto e^{2\pi i t}$, which is a homomorphism $(\mathbb R,+) \to (\mathbb C^*,\times)$ whose kernel is $\mathbb Z$ and whose image is the unit circle.
In this context, it is instructive to consider what $\mathbb{Q}/\mathbb{Z}$ is.
A: $ \mathbb{R}/\mathbb{Z} $ doesn't make sense when viewed as a quotient ring, because $\mathbb{Z} $ is not an ideal of $ \mathbb{R} $, so I will view it as a quotient group.
In that case, the quotient group $\mathbb{R}/\mathbb{Z}$ consists of shifted copies of $\mathbb{Z}$ by each element of $[0, 1)$, and is an uncountable set. More explicitly, we have that $ \mathbb{R}/\mathbb{Z} = \{ \mathbb{Z} + r : r \in [0, 1) \} $, and it is easy to check that all of these cosets are distinct. You can think of this as defining an equivalence relation $ x \equiv y $ on $\mathbb{R}$ such that $ x \equiv y $ iff $ x - y \in \mathbb{Z} $, and then considering the set of all equivalence classes of this relation.
A: By definition, two elements $x, y \in \Bbb R$ are in the same equivalence class (group coset under $+$) in $\Bbb R / \Bbb Z$ iff $y - x \in \Bbb Z$, that is, if $x$ and $y$ have the same fractional part. So, just as every element in $\Bbb Z / n \Bbb Z$ has a unique representative in $\{0, \ldots, n - 1\} \subset \Bbb Z$, every element of $\Bbb R / \Bbb Z$ has a unique representative in, e.g., $[0, 1)$.
Edit Re the additions to the question. It's true that the notation $A / B$ is overloaded, but in each case one assumes that the notion of quotient is appropriate to the context (or put a little more precisely, the category of the objects $A, B$): If $A$ and $B$ are suitable topological spaces, one assumes by default that this is the topological quotient space, whereas if they are appropriate algebraic objects, one assumes that we're working with a notion of quotient that respects the algebraic structure (e.g., a notion of cosets in a gorup).
For example, the cone construction $X \rightsquigarrow (X \times I) / (X \times \{0\})$ is an essentially topological one---indeed, what other quotient structure is available (besides the underlying set one)? On the other hand, as in the example $\Bbb R / \Bbb Z$, the objects have both a usual group structure and a usual topology, and it's not explicit which notion of quotient is being used. In this case, the group quotient map $\Bbb R \Bbb R / \Bbb Z$ occurs all the time in the wild (surely it is the most important example of a covering map of Lie groups). The topological quotient of $\Bbb R$ formed by identifying all of the integers defines a "bouquet of countably many circles", but in practice it occurs vanishingly less commonly that the above algebraic notion. That said, if you want your writing to be totally unambiguous, specify which you mean!
A: The "slash" notation is used for various different kinds of quotient objects, with varying meanings depending on the context or category. It can be confusing to keep them all straight.
There are actually two or three competing meanings of the "slash" in the expression $\mathbb{R}/\mathbb{Z}$.
First, there is the "group theoretic" meaning: for any group $G$ and any normal subgroup $N$, $G/N$ denotes the quotient group. One way to identify the quotient group $Q$ up to isomorphism is to make an educated guess at what $Q$ might be, then to construct a surjective homomorphism $G \mapsto Q$ with kernel $N$. As other answers have indicated, $\mathbb{R}/N$ is isomorphic to the "circle group" $S^1$ consisting of complex numbers with unit norm, under the operation of complex multiplication, and the surjective homomrophism $\mathbb{R} \mapsto S^1$ one constructs for the proof is the map $t \mapsto e^{2\pi i t}$.
Second, there is the "topological group action" meaning: for any topological space $Y$ and any action of a group $H$ on $Y$, $Y/H$ denotes the orbit space, defined as follows. The orbits are defined to be the subsets of the form $O_y = \{h \cdot y \,\bigm|\, h \in H\}$. These subsets form a partition of $Y$, and $Y/H$ is the quotient space of this patition. One way to identify the orbit space $Y/H$ up to homeomorphism is to make an educated guess at a topological space $Q$ that might be homeomorphic to $Y/H$, and then to construct a quotient map $f : Y \mapsto Q$ such that the set of point preimages $\{f^{-1}(q) \,\bigm|\, q \in Q\}$ coincides with the set of orbits $\{O_y \,\bigm|\, y \in Y\}$. In your example, where $Y=\mathbb{R}$ and $\mathbb{Z}$ acts on $\mathbb{R}$ by addition, once again $Q=S^1$ and $f : \mathbb{R} \to S^1$ is the map $f(t)=e^{2\pi i t}$.
Third, there is the "topological group" meaning: $\mathbb{R}$ is a topological group, $\mathbb{Z}$ is a normal subgroup, and $\mathbb{R}/N$ is the quotient topological group. Once again, one can formalize what "quotient" means in the context of topological groups and normal subgroups; intuitively this means that it is simultaneously a group theoretic quotient and a topological group action quotient. Once again one can prove that $\mathbb{R}/\mathbb{Z}$ is isomorphic, in the category of topological groups, to the topological group $S^1$.
Also, the "slash" is used with a "topological quotient space" meaning: given a topological space $X$ and an equivalence relation denoted $\sim$, sometimes one uses the expression $Y/\!\sim$ to denote the quotient space defined by the decomposition of $Y$ into equivalence classes of $\sim$. The "topological group action" quotient discussed above is a special kind of topological quotient space, where the equivalence relation $\sim$ is defined by $y \sim y'$ if and only if $O_y=O_{y'}$.
