Difference between $SU(2)$ and $SU(2)$ gauge transformations? I hear this jargon all the time, so what is the difference?
(Of course this is nothing special to $SU(2)$, but rather I just took it as an example)
 A: $SU(2)$ is a finite-dimensional Lie group.  The elements of this group are $2 \times 2$ matrices $A$ which are unitary ($A^\dagger = A^{-1}$) and have determinant $1$.  More generally, $G$ may be a reductive Lie group.  Physicists usually prefer to talk about products of $U(1)$ and semisimple groups.
A gauge transformation is a function from spacetime $\Sigma$ into the group $G$.   This is a map $g$ which assigns to every $x \in \Sigma$ a matrix $g(x) \in SU(2)$.  ('$SU(2)$ gauge transformation' denotes the special case $G=SU(2)$.)  The set $\mathcal{G} = Map(\Sigma,G)$ of all maps $g: \Sigma \to G$ is a group because we can take two maps $g$ and $h$ and define a new map $gh$ via $gh(x) = g(x)h(x)$.  The identity is the constant map given by $e(x) = 1$, where $1$ is the identity in $G$.
When being precise, physicists tend to reserve the term 'gauge transformation' for gauge transformations which belong $\mathcal{G}_0$, the connected component of the identity in $\mathcal{G}$;  these are points in $\mathcal{G}$ which are connected to the identity by a continuous path.  (It is considered impolite on physics boards to pay too much attention to what the topology is on $\mathcal{G}$ and exactly how well-behaved the maps ought to be.  However, for lattice gauge theory, you really want to ensure that you can approximate any gauge transformation well by a map from the lattice to $G$.)
The reason for doing this is that physicists don't usually want to allow constant functions (or, at least, constant and valued in the identity component of $G$) to count as gauge symmetries.  Gauge symmetries are supposed to leave the physical system invariant.  But in electromagnetism, for example, we want the global group $U(1)$ to be a Noether group, whose conserved charge is electric charge.  Likewise, in QCD, the statement that only color singlets are observable at large distances is meant to have physical content.  Confinement is not a triviality.  
Sometimes people use the terminology 'global gauge transformation' to refer to such constant functions to $G$.  This is perverse terminology, but good luck getting a newspaper to publish a letter to the editor complaining about this problem.
A: I am not an expert but lets give this a shot. I really recommend reading up on classical theory of gauge fields by Rubakov, first; Its a life saver. All I have below is just mostly words, but the book I recommend is great.
There are a few terms in the business :
Gauge group: Some group of principal bundles on which a field is connected. eg $SU(N)$
Gauge transformation: $A \rightarrow A- \nabla \psi$
and                    $\phi \rightarrow \phi + a$
Gauge Invariance: Your New Lagrangian $L \rightarrow L'$ does not change the theory
Also consult comments  @Hunter's comment below
Some other things:
You will surely come across two motivated responses to your questions, A mathematical one, and a physical one, I recommend spending time on both
I have to again say, I am not any sort of expert, but I hope this helps. I am open to any suggestions on how to edit my response.
A: SU(2) is a type of group, that is a collection of objects that have certain properties.
An SU(2) gauge transformation refers to the observation that certain objects in certain equations remain invariant if you multiply them by the elements of the SU(2) group.
How this works is easiest to see with rotations on vector dot products.  Consider an equation that only has terms like $\vec{v} \cdot \vec{v}$.  Now let us rotate our coordinate system by some matrix $R$ (in this case $R$ belongs to the group SO(3)).  Then, 
$\vec{v} \cdot \vec{v} = v^T v -> v^T R^T R v = v^T v$
since $R^T R = 1$ is one of the defining properties of the group SO(3).
To recap, SU(2) in your example refers to a group of objects and an SU(2) gauge transformation refers to the act of multiplying objects in an equation by the objects in the group SU(2) and noting that the equation doesn't change under such multiplications (just like the length of a vector doesn't change under rotations).
