  This article gives an exposition of the deformation theory for pairs $(X,
E)$, where $X$ is a compact complex manifold and $E$ is a holomorphic vector
bundle over $X$, adapting an analytic viewpoint \`{a} la Kodaira-Spencer. By
introducing and exploiting an auxiliary differential operator, we derive the
Maurer--Cartan equation and differential graded Lie algebra (DGLA) governing
the deformation problem, and express them in terms of differential-geometric
notions such as the connection and curvature of $E$, obtaining a chain level
refinement of the classical results that the tangent space and obstruction
space of the moduli problem are respectively given by the first and second
cohomology groups of the Atiyah extension of $E$ over $X$. As an application,
we give examples where deformations of pairs are unobstructed.
