In modern mathematical physics, one is often times concerned with the equations of motion of a certain class of physically representative objects. In field theory over a curved spacetime, these typically take the form of some certain systems of PDEs, such as the Dirac equation (for the electron-positron field). One notices these equations are formulated in terms of spatiotemporal derivatives of matter fields, which (technically speaking) are sections of so-called associated vector bundles to principal bundles. It turns out that the data of a smooth principal connection, in classical smooth differential geometry, is exactly the data needed to systematically formulate these spatiotemporal derivations of sections of associated vector bundles. The coefficients of the principal connection are identified with gauge fields such as the electromagnetic \(4\)-potential. Deep as it may be, the entire framework, in its full glory, is undeniably intricate; we seek to tame this tendency by judicious application of algebra. Moreover, once this algebraic approach is in hand, there is no real issue (in principal) if we were to replacing our smooth algebras with more general algebras, even noncommutative. Indeed, a popular approach in the literature both now and from the past few decades is to use this framework to formulate an algebraic-geometric theory of quantum mechanics by using this setup with non-commutative algebras which one encounters in quantum mechanics. This thesis develops the framework, and then takes it for a test run on some examples which are familiar to most working mathematicians. In particular, the final few examples are complete classifications of noncommutative principal connections on (in reality commutative) noncommutative principal bundles; it is through these examples that we see our noncommutative theory even still sheds new insight on the classical commutative theory by generalizing the breadth of geometrically meaningful algebras the theory can cope with.