  In this paper, we introduce the discrete conformal structures on surfaces
with boundary in an axiomatic approach, which ensures that the Poincar\'{e}
dual of an ideally triangulated surface with boundary has a good geometric
structure. Then we classify the discrete conformal structures on surfaces with
boundary, which turns out to unify and generalize Guo-Luo's generalized circle
packings \cite{GL2}, Guo's vertex scalings \cite{Guo} and Xu's partial discrete
conformal structures \cite{Xu22} on surfaces with boundary. Motivated by
\cite{BPS, Zhang-Guo-Zeng-Luo-Yau-Gu}, we further study the relationships
between the discrete conformal structures on surfaces with boundary and the
hyperbolic trigonometry. Unexpectedly, we find that some subclasses of the
discrete conformal structures on surfaces with boundary are closely related to
the twisted generalized hyperbolic triangles introduced by Roger-Yang
\cite{Roger-Yang}, which does not appear in the case of closed surfaces.
Finally, we study the relationships between the discrete conformal structures
on surfaces with boundary and the 3-dimensional hyperbolic geometry by
constructing ten types of generalized hyperbolic tetrahedra. Some new
generalized hyperbolic tetrahedra recently introduced by Belletti-Yang
\cite{B-Y} naturally appears in the constructions.
