It was very surprising when two cohorts, namely mathematicians and physicists, found that the stuff they were doing simultaneously was the same in two different languages. Let us take a vector bundle E with $G$ as its structure group on a manifold $\mathcal{M}$. Then, here, one can do lots of differential geometry. But the interesting thing is that this is equivalent to describing a gauge theory with the gauge group $G$. Let's say the connection on the vector bundle $A$ is 1-forms potential in the gauge theory. The curvature is defined as the exterior derivative of $A$

$$F=dA$$
In the case of electromagnetism, which is U(1) gauge theory, one can say that a vector bundle with $U(1)$ structure group and connection $A$ defines the electromagnetism where $A$ follows a gauge transformation

$$G \colon A \rightarrow A'$$

where $G=U(1)$ in this case. The gauge transformations are taken in the overlap regions which we define while defining the 1-forms potentials (which are not global on the manifold). The curvature here is the 2-form $F$, which is the field for electromagnetism. It follows a Bianchi identity

$$dF=0.$$

Similarly, we can describe gauge theory using this way of seeing for $G=SU(2)$ or $G=SO(3)$. However, every case has its own special features; for example, if one has a compact abelian group (like U(1)), one can shoot for Hodge theory to study the 2-forms.

Now, the gauge group $G$ is the automorphism (the homomorphism from $E$ to $E$) of the bundle, and the algebra associated with $G$ is called gauge algebra. Thus, we observe that the Yang-Mills theory is described by these connections, curvature, and gauge group of a vector bundle.

We will expand later on why the connections are defined only locally on a manifold. For reference, one may check this book or this. There exists a very giant literature (expository too) on this.