Yang-Mills potential and principal bundles
The term gauge transformation refers to two related notions in this context. Let $P$ be a principal $G$-bundle over a manifold $M$, and let $\cup_i U_i$ be a cover of $M$. A connection on $P$ is specified by a collection of $\mathfrak{g}=\mathrm{Lie}(G)$ valued 1-forms $\{A_i\}$ defined in each patch $\{U_i\}$, together with $G$-valued functions $g_{ij} : U_i \cap U_j \to G$ on each double overlap, such that overlapping gauge fields are related by
$$A_j = g_{ij} A_i g_{ij}^{-1} + g_{ij} \mathrm{d} g_{ij}^{-1}.\tag{1}$$
The transition functions must also satisfy the cocycle condition on triple overlaps, $g_{ij}g_{jk}g_{ki} =1$. This is the first notion of a gauge transformation, relating local gauge fields on overlapping charts.
Second, there is a notion of gauge equivalence on the space of connections. Two connections $\{ A_i, g_{ij} \}$ and $\{A_i',g_{ij}'\}$ are called gauge-equivalent if there exist $G$-valued functions $h_i : U_i \to G$ defined on each patch such that $$A_i' = h_i A_i h_i^{-1} + h_i \mathrm{d}h_i^{-1} ~~\text{and}~~ g_{ij}' = h_j g_{ij} h_i^{-1}\tag{2}$$
In terms of the globally defined connection 1-form $\omega$ on $P$, the local gauge fields $\{A_i\}$ are defined by choosing a collection of sections $\{\sigma_i\}$ on each patch of $M$. The local gauge fields are obtained by pulling back the global 1-form, $A_i = \sigma_i^* \omega$. On overlapping patches, such pullbacks are related by (1). On the other hand, the choice of sections was arbitrary; a different collection of sections $\{\sigma'_i\}$ related to the first by $\sigma'_i = \sigma_i h_i$ leads to the gauge-equivalence (2).
Given a map $f: M \to M'$ between two manifolds and a bundle $P'$ over $M'$, we obtain a bundle over $M$ by pullback, $f^* P'$. Moreover, the pullback bundle depends only on the homotopy class of $f$. Suppose we have a contractible manifold $X$. By definition, there exists a homotopy between the identity map $\mathbf{1}:X \to X$ and the trivial map $p: X \to X$ which takes the entire manifold to a single point $p\in X$. Let $P$ be a bundle over $X$. The identity pullback of course defines the same bundle, $\mathbf{1}^* P = P$. On the other hand, the pullback $p^* P$ is a trivial bundle; it maps the same fiber above $p$ to every point on $X$. But the bundles $\mathbf{1}^*P$ and $p^*P$ are equivalent since $\mathbf{1}$ and $p$ are homotopic maps. Thus, a bundle over a contractible space is necessarily trivial (i.e. a direct product).
In particular, a $G$-bundle over $\mathbb{R}^4$ is trivial, whether $G$ is abelian or non-abelian. The cover $\cup_i U_i$ has a single chart, $\mathbb{R}^4$ itself. There is a single gauge field $A$, which is a globally defined $\mathfrak{g}$-valued 1-form. It is obtained from the 1-form $\omega$ on $P$ by pullback, $A = \sigma^* \omega$, where $\sigma$ is a globally defined section. Picking another section $\sigma' = \sigma g(x)$ produces a gauge-equivalent connection, related to $A$ by the usual gauge transformation law given above.
For more details, see e.g. Nakahara "Topology, Geometry, and Physics," chapter 10.
There is something a little more subtle happening even in the case where spacetime is contractible $\mathbb{R}^4$. Even in this trivial setting, it is usually required (for physical reasons) that the connection one form decays at infinite radius. That is $$A(x) \to 0\text{ as } |x|\to\infty.$$ As user81003's answer has already mentioned, we can only determine the connection 1-form up to gauge equivalence, so the condition that $A(x) \to 0$ is too strong. The best we can do is require that $$A(x) \to h(x) d h(x)^{-1}\text{ as }|x|\to\infty.$$ for some choice of $G$-valued function $h(x)$ (which may only be defined for $|x|$ sufficiently large).
There are two ways to interpret how this leads to topologically non-trivial bundles. The simpler (and more heuristic) way is to say that this decay condition means choosing the function $h(x) : S^3 \to G$ on the sphere at infinity. This "boundary condition" should only be defined up to continuous deformation, so we really only care about the homotopy class of $h$, which is an element of $\pi_3(G)$.
An alternative view is to observe that the condition that $A(x)$ is gauge-equivalent to $0$ at infinity means that we may add the point $\infty$ to our spacetime manifold and the connection $A$ will still be well defined up to gauge. This means we should really look at principal $G$ bundles on the compactification of $\mathbb{R}^4$, which is $S^4$. Since $S^4$ is not contractible, it is no longer true that all principal $G$ bundles are trivial. As explained in user81003's answer, we now need to choose trivializing charts of $S^4$, which we can take to be the disks corresponding the the northern and southern hemispheres of the sphere (each extended a bit so they overlap). The intersection of these charts is the equator times a small interval $S^3\times (-\epsilon, \epsilon)$, which is homotopy equivalent to $S^3$. The transition function on this overlap $$g_{12} : U_1\cap U_2 \simeq S^3 \to G$$ then classifies the bundle. Again, this is only defined up to continuous deformation, so we see that the class of the principal $G$ bundle is determined by an element of $\pi_3(G)$.
At this point, it is clear that the gauge group makes a difference as to whether there can be nontrivial bundles. For instance, since $U(1) \cong S^1$, we see that every map $S^3 \to S^1$ is homotopic to the constant map (in other words, $\pi_3(S^1) = 0$), so there are still no non-trivial $U(1)$ bundles on $\mathbb{R}^4$, even with the decay condition taken into account (as user81003 said, this does not mean there are no gauge transformations!). However, if $G = SU(2) \cong S^3$, then we see that $\pi_3(S^3) \cong \mathbb{Z}$ (the integer counts the degree of the map $S^3\to S^3$), so there are some nontrivial possibilities. This gives a topological interpretation for $SU(2)$ instantons on $\mathbb{R}^4$. There is a very brief discussion of this at https://en.wikipedia.org/wiki/Instanton#Quantum_field_theory but perhaps others know of better references.