What does a *pair* mean in the definition of a topological space? In the definition of a topological space, a topological space $(X,\tau)$ is a a pair where $X$ is the set on which the topology is defined and $\tau$ is a topology which is a subset of $\mathcal{P}(X)$ defined with specific conditions. 
What I don't understand is what exactly is meant by the pair $(X,\tau)$. 
What does an element from this "topological space" look like? Is it also a pair?
Is this supposed to be a tuple, since that is what the $(\cdot,\cdot)$ suggests?
Is there a way I can understand this geometrically, e.g. what would $(\mathbb{R},\tau)$ look like? 
I want to understand how I can envision a topological space.
 A: 
What I don't understand is what exactly is meant by the pair $(X,\tau)$.

Pair is a pair, two things in correct order. That's it. Seriously. It's like bicycle is a pair of wheels together with some skeleton and other mechanisms. Of course in mathematics pair has concrete definition, but it is not really relevant. What matters is that pair is an ordered collection of two things.
So a topological space is a set together with some additional structure, called topology. The reason we say "a topological space is a pair $(X,\tau)$ such that..." is because the topological space depends on both the underlying set $X$ and the topology $\tau$. I.e. on a given set $X$ we can define multiple different topological structures, e.g. $\tau=\{\emptyset, X\}$ (a.k.a. antidiscrete topology) and $\tau=P(X)$ (a.k.a. discrete topology) are two different (unless $|X|\leq 1$) topologies on $X$.

What does an element from this "topological space" look like? Is it also a pair?

Strictly, formally speaking, we never talk about elements of a topological space (since formally any topological space $(X,\tau)$ has exactly two elements: $X$ and $\tau$). When we say that $x$ is an element of a topological space $(X,\tau)$ then typically what we mean is that $x\in X$, i.e. we only talk about elements of the underlying set.

Is there a way I can understand this geometrically, e.g. what would $(\mathbb{R},\tau)$ look like?

That depends on $\tau$. When $\tau$ is the Euclidean topology, then we deal with the standard Euclidean line. But there are infinitely many nonequivalent topologies $\tau$ on $\mathbb{R}$. Some of them easy to understand, e.g. discrete topology on $\mathbb{R}$ makes every point isolated. But some of them are quite abstract and weird to wrap mind around, e.g. high dimensional spaces (note that since $\mathbb{R}$ is equinumerous with any $\mathbb{R}^n$ then for any $n$ we can define topology on $\mathbb{R}$ making it $n$-th dimensional).
A: This is just common mathematical language. No strange mistery hidden iside.
When you speak about a "topologial space" you mean just a set $X$, but on which is given a topology $\tau$. The fact that you have a topology on $X$ does not change the fact that $X$ is just a set.
So, the elements of a topological space are just its elements as a set.
A topological space IS A SET. Equipped with a topology.
So, mathematicians, when have to say that in a short formula, say $(X,\tau)$.  But every time you read "let $(X,\tau)$ be a topological space" you can replace it with "let $X$ be a set, and let $\tau$ be a topology on $X$". The two sentences are perfect synonimous.
A: A pair is just a 2-tuple, to be said, an ordered set of two elements. In topology, the definition of a topological needs two things: a set and a topology. This means that the only way to correctly determine a topological space, both things need to be determined, using the notation $(X,\tau)$. It's important to define it as a pair, because the same set with two different topologies are two distinct topological spaces; and two different sets with the same topology are again two completely different topological spaces.
Despite this, it's common to see notation abuses like naming $(X,\tau)$ just $X$, but that requires to specify first what $\tau$ you're using, and for it's elements it's just correct to say $x\in (X,\tau)$.
A: A pair is similar to a set of two elements, with the difference that the order matters. A topological space $(X, \tau)$ is defined by a set ($X$) and a topology ($\tau$). So we need both to appear somewhere.
However, they don't play the same role and are not interchangeable. That's why we use $(X, \tau)$ rather than $\{X, \tau \}$. Put simply, as you probably already know (since you mention tuples), if $a \neq b$, then:
$\{a,b\} = \{b,a\}$
$(a,b) \neq (b,a)$
An element of a topological space $(X, \tau)$ is just an element of $X$.  $(\mathbb{R},\tau)$ is just $\mathbb{R}$ where a specific collection of subsets are chosen to be "open sets". I don't think there's a way to visualize all of them at once. But we must include $\tau$ in the definition because the topological properties of $(\mathbb{R}, \tau)$ only make sense for that specific topology $\tau$
