Let $\sigma_i$ be the Pauli matrices.
A typical representation of $\mathfrak{su}(2)$ is $i\sigma_i$.
Any 3-vector $x$ can be represented as
$$X = \sigma\cdot x,$$
where $\sigma\cdot x = \sum_{i=1}^3 \sigma_i x_i$.
($i X$ is a pure imaginary quaternion.)
The components of $x$ can be found from $X$, using the fact that
$\mathrm{Tr}\, \sigma_i \sigma_j = 2\delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta.
We have
$$x_i = \frac{1}{2} \mathrm{Tr}\, \sigma_i X.$$
Rotate the vector $x$ about the axis $n$ by the angle $\theta$.
The rotated vector $x'$ is represented by
$$\begin{equation*}
X' = R^{-1} X R \tag{1}
\end{equation*}$$
where
$$R =
e^{i\theta n\cdot \sigma/2}
= \mathbb{I} \cos\frac{\theta}{2} + i n\cdot\sigma \sin \frac{\theta}{2}.$$
(In the language of quaternions, $R$ is a versor.)
Note that $R^{-1}(\theta) = R(-\theta)$.
Equation (1) makes the fact that $\mathrm{SU}(2)$ is the double cover of $\mathrm{SO}(3)$ explicit---in $\mathrm{SU}(2)$ $R(\theta+2\pi) = -R(\theta)$, but in $\mathrm{SO}(3)$ these rotations are indistinguishable.
The fact that (1) represents the appropriate rotation in 3-space can be proved by showing, for example, that it gives Rodrigues' rotation formula, relating $x'$ to $x$ in the appropriate way.
Roughly, $X$ transforms like a vector, that is, like the $(1/2,1/2)$ representation of $\mathrm{SU}(2)$---its left index transforms with $R^{-1}$ and its right index with $R$.
It is a good exercise to show using this formalism that the vector $x$ rotated about the $z$-axis by the angle $\theta$ is
$$x' =
\left(\begin{array}{c}
x_1 \cos\theta - x_2\sin\theta \\
x_1 \sin\theta + x_2\cos\theta \\
x_3
\end{array}\right)
= \left(\begin{array}{ccc}
\cos\theta & -\sin\theta & 0 \\
\sin\theta & \cos\theta & 0 \\
0 & 0 & 1
\end{array}\right)
\left(\begin{array}{c}
x_1 \\
x_2 \\
x_3
\end{array}\right)$$
as required.
![Rotating gimbal-xyz.gif, By Lars H. Rohwedder (User:RokerHRO) (Own work) [Public domain], via Wikimedia Commons](https://i.stack.imgur.com/tEJA4.gif)
Figure 1. Gimbal lock.
As mentioned in the interesting link provided by @JyrkiLahtonen in the comments, gimbal lock can affect gimbals as well as mathematical representations of rotations.
Rotations by Euler angles suffer this problem.
Let $R(\alpha,\beta,\gamma)$ be the usual representation of a rotation in 3-space by Euler angles.
If $\beta = 0$, then $R(\alpha,0,\gamma) = R(\alpha+\gamma,0,0) = R(0,0,\alpha+\gamma)$.
We have lost a degree of freedom---we can only rotate about the $z$-axis.
$\mathrm{SU}(2)$ representations of rotations in 3-space don't have this problem.
For example, no matter how we choose $\theta$, we will not lose any degrees of freedom in $n$ due to that choice.
If you are familiar with topology, these statements are a consequence of the fact that the three torus is not a covering space of $\mathrm{SO}(3)$, but that $\mathrm{Spin}(3) = \mathrm{SU}(2)$ is its universal cover.
I recommend using Wikipedia and Google to learn more about this interesting subject.