Often, the definition of Riemann surfaces is motivated by the example of the multi-valued function $f(z)=\sqrt{z}$. Every point $z\in \Bbb{C}$ has two images. Hence, this function has two "branches"; $re^{i\theta}\to \sqrt{r}e^{i\theta/2}$ and $re^{i\theta}\to \sqrt{r}e^{i(\theta+ 2\pi)/2}$. Here, $0\leq\theta<2\pi$ and $r\geq 0$.
We choose one of these two branches.
Why do we then get two copies of the complex plane, and the consequent gluing, etc? Now that we have already chosen one branch, shouldn't we be satisfied with whatever images we get (all of which lie in the upper half plane, as in our case)?
The point is that we'd like to define a function "sorta like square root" with the following property:
If $\gamma: [a, b] \to \mathbb C - \{0\}$ is a curve, then
$F(\gamma(t))^2 = \gamma(t)$ (so $F(z)$ is a square root of $z$), and
$t \mapsto F(\gamma(t))$ is continuous as a function of $t$.
In other words, the proposed function $F$ would let us compute square roots "in a continuous way depending on position."
Alas, that's not possible for any function $F: \mathbb C \to \mathbb C$, for if $\gamma$ traverses the unit circle once, we find that $F(1)$ must be both $+1$ and $-1$.
On the other hand, we can build a surface $M$ with the property that
There's a function $p: M \to \mathbb C$ that's smooth and is 2-to-1 almost everywhere (i.e., for each point of $\mathbb C$, there are two points of $M$), the exception being that $p^{-1}(0)$ consists of a single point of $M$.
every continuous path in $\mathbb C - \{0]\}$ has exactly two "lifts" to $M$, i.e., for a such a path $\gamma$, there's a continuous path $\alpha: [a,b] \to M$ with $p(\alpha(t)) = \gamma(t)$ for every $t$. (And in fact there are two of these).
And on this "Riemann surface" $M$, there is a function like the one we were looking for, i.e., there's a continuous function $$ F: M \to \mathbb C $$ with the property that if $\alpha$ is a lift of a curve $\gamma$ that misses the origin, then $$ F(\alpha(t))^2 = \gamma(t) $$ for every $t$. So on this Riemann surface, "there are square roots".