Let $\mathcal{H}_j$ be the space of polynomial functions on $\mathbb{C}^2$ that are homogemeous of degree $2j$. An element in $\mathcal{H}_j$  is a polynomial in complex variables $x$ and $y$ that is a linear combination of polynomials $x^py^q$ where $p+q=2j$. $\mathcal{H}_j$ has dimension $2j+1$ since it has a basis given by
$$x^{2j},x^{2j-1}y,x^{2j-2}y^2,\cdots,y^{2j}.$$
For any $g\in\mathrm{SU}(2)$, let $U_j(g)$ be the linear transformation of $\mathcal{H}_j$ given by
$$U_j(g)f(v)=f(g^{-1}v)$$
for $f\in\mathcal{H}_j$ and $v\in\mathbb{C}^2$. Then $U_j$ is a representation: $U_j(I)$ is the identity. For any $g,h\in\mathrm{SU}(2)$,
\begin{align*}
U_j(g)U_j(h)f(v)&=U_(h)f(g^{-1}v)\\
&=f(h^{-1}g^{-1}v)\\
&=f((gh)^{-1}v)\\
&=U_j(gh)f(v)
\end{align*}
for $f\in\mathcal{H}_j$ and $v\in\mathbb{C}^2$.

Physicists call $U_j$ spin-$j$ representation. Since $2j+1$ has to be a positive integer, we have spin-0 representation, spin-$\frac{1}{2}$ representation, spin-1 representation, etc. It is interesting to see the correspondence between spin-$j$ representation and particles. The only known spin-0 particle is Higgs-boson, the so-called God particle, which is responsible for giving masses to bosons. The Higgs-boson appears to have been discovered recently by the LHC (Large Hadron Collider) at CERN. Spin-$\frac{1}{2}$ particles are fermions which include all quarks and leptons. Spin-1 particles are gauge bosons (force-carrying particles) such as photons, W bosons, Z bosons, gluons. Curiously there are currently no spin-$\frac{3}{2}$ particles predicted in particle physics. The hypothetical gravitons are believed to be spin-$2$ particles.

Proposition. The spin-0 representation of $\mathrm{SU}(2)$ is equivalent to the trivial representation in which every element of the group acts on $\mathbb{C}$ as the identity.

Proposition. The spin-$\frac{1}{2}$ representation of $\mathrm{SU}(2)$ is equivalent to the fundamental representation in which every element $g\in\mathrm{SU}(2)$ acts on $\mathbb{C}^2$ by matrix multiplication.

Note that the $U_j$ are irreducible and that they are all of the irreducible representations.

$\mathbb{R}^3$ can be identified with the set of $2\times 2$ Hermitian matricies of the form
\begin{align*}
X&=\begin{pmatrix}
z & x-iy\\
x+iy & -z
\end{pmatrix}\\
&=z\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}+x\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}+y\begin{pmatrix}
0 & -i\\
i & 0
\end{pmatrix}\\
&=x\sigma_1+y\sigma_2+z\sigma_3,
\end{align*}
where $\sigma_1,\sigma_2,\sigma_3$ are called the Pauli spin matrices in physics. Define an inner product $\langle\ ,\ \rangle$ on the Hermitian matrices by
$$\langle X, Y\rangle=\frac{1}{2}\mathrm{tr}(XY).$$
In particular,
$$|X|^2=\frac{1}{2}\mathrm{tr}(X^2)=-\det X.$$
With this inner product, the identification is an isometry. $\mathrm{SU}(2)$ acts on $\mathbb{R}^3$ via the representation
$$\rho:\mathrm{SU}(2)\longrightarrow\mathrm{GL}(3,\mathbb{R})$$
defined by
$$\rho(U)X=UXU^{-1}$$
for $U\in\mathrm{SU}(2)$ and $X\in\mathbb{R}^3$. It turns out that $X\longmapsto UXU^{-1}$ is an orientation preserving isometry of $\mathbb{R}^3$, so
$$\rho:\mathrm{SU}(2)\longrightarrow\mathrm{SO}(3).$$
Since both $U$ and $-U$ result the same isometry, the representation $\rho:\mathrm{SU}(2)\longrightarrow\mathrm{SO}(3)$ is a $2:1$ map. Since $\mathrm{SU}(2)=S^3$ is simply-connected, $\rho$ is a universal convering map and that we have
$$\mathrm{SU}(2)/\mathbb{Z}_2=\mathrm{SO}(3).$$
The quotient group $\mathrm{SU}(2)/\mathbb{Z}_2$ is denoted by $\mathrm{PSU}(2)$ and called the projective special unitary group.

The double cover i.e. $2:1$ cover of $\mathrm{SO}(n)$ is called the spin group and is denoted by $\mathrm{Spin}(n)$. For $n>2$, $\mathrm{Spin}(n)$ is simply-connected so it is the universal cover of $\mathrm{SO}(n)$. Some examples of spin groups are
\begin{align*}
\mathrm{Spin}(1)&=\mathrm{O}(1)=\mathbb{Z}_2=\{\pm I\}\\
\mathrm{Spin}(2)&=\mathrm{U}(1)=\mathrm{SO}(2)\\
\mathrm{Spin}(3)&=\mathrm{SU}(2)\\
\mathrm{Spin}(4)&=\mathrm{SU}(2)\times\mathrm{SU}(2)
\end{align*}
Note that $\mathrm{SO}(3)\subset\mathrm{GL}(3,\mathbb{R})\subset\mathrm{GL}(3,\mathbb{C})$, so for any $g\in\mathrm{SU}(2)$, $\rho(g):\mathbb{C}^3\longrightarrow\mathbb{C}^3$. Hence, $\rho$ is in fact equivalent to the spin-1 representation of $\mathrm{SU}(2)$.

In quantum mechanics, unitary representation is particularly important. Let $\mathcal{H}$ be the Hilbert space of states derived from a quantum mechanical system. Let $\rho$ be a representation of a Lie group $G$ on $\mathcal{H}$ i.e. $\rho:G\longrightarrow\mathrm{GL}(\mathcal{H})$. $\rho$ is called a unitary representation if
$$\langle\rho(g)\psi,\rho(g)\phi\rangle=\langle\psi,\phi\rangle$$
for all $g\in G$ and $\psi,\phi\in\mathcal{H}$. Intuitively each $\rho(g)$ may be understood as a rotation. For example, say $G=\mathrm{SO}(3)$. First rotating the particle by some amount $h\in\mathrm{SO}(3)$ and then rotating it by some amount $g\in\mathrm{SO}(3)$ should have the same effect as rotating it by the amount $gh\in\mathrm{SO}(3)$. That is $\rho(g)\rho(h)=\rho(gh)$. This tells why we need a representation in quantum mechanics. In quantum mechanics, the inner product $\langle\ ,\ \rangle$ measures probability. For instance if a particle is in the state $\psi$, then $\langle\psi,\phi\rangle$ is the probability of finding the particle in the state $\phi$. Rotating a particle amounts to a change of coordinates and the state of a particle should not depend on a change of coordinates. Hence, in quantum mechanics we require representation to be unitary.

Lastly I would like to mention the difference between bosons and fermions in terms of representation. Let $f\in\mathcal{H}_j$, the spin-$j$ representation space. Then
$$U_j(-I)f(v)=f(-v)=(-1)^{2j}f(v)$$
since $f$ is a homogeneous polynomial of degree $2j$. This implies that
$$U_j(-I)=(-1)^{2j}.$$
Hence, $U_j$ maps both $I$ and $-I$ to the identity if $j$ is an integer, while it does not if $j$ is a half-integer.

References:

[1] John Baez, Javier P. Muniain, Gauge Fields, Knots and Gravity, World Scientific 1994

[2] Brian C. Hall, Lie Groups, Lie Algebras, and Representations, An Elementary Introduction, Springer-Verlag 2003