We prove a structural theorem for unit-disk graphs, which (roughly) states that given a set of unit disks inducing a unit-disk graph and a number , one can partition into subsets such that for every and every , the graph obtained from by contracting all edges between the vertices in admits a tree decomposition in which each bag consists of cliques. Our theorem can be viewed as an analog for unit-disk graphs of the structural theorems for planar graphs and almost-embeddable graphs proved recently by Marx et al. [SODA ’22] and Bandyapadhyay et al. [SODA ’22]. By applying our structural theorem, we give several new combinatorial and algorithmic results for unit-disk graphs. On the combinatorial side, we obtain the first Contraction Decomposition Theorem for unit-disk graphs, resolving an open question in the work by Panolan et al. [SODA ’19]. On the algorithmic side, we obtain a new algorithm for bipartization (also known as odd cycle transversal) on unit-disk graphs, which runs in time, where denotes the solution size. Our algorithm significantly improves the previous slightly subexponential-time algorithm given by Lokshtanov et al. [SODA ’22] which runs in time. We also show that the problem cannot be solved in time assuming the Exponential Time Hypothesis, which implies that our algorithm is almost optimal.