Gauss-Bonnet says closed hyperbolic surfaces have negative Euler characteristic, and so we immediately see that a torus (without boundary or punctures) cannot be equipped with a hyperbolic structure.
However, if when we add punctures or boundary to the torus it seems that we can endow it with a hyperbolic metric? Is it true that all bordered surfaces admit hyperbolic metrics?
By bordered surface I mean that the surface has punctures and/or boundary with marked points on each boundary component.
Yes, all bordered surfaces $S$ admit hyperbolic metrics. Here's a construction which reduces the question to one about surfaces with empty boundary, where the question has a simpler answer.
Start with $DS$ which is the double of $S$ across its boundary, namely the union of two copies of $S$ where one identifies each point pair on the boundaries of these two copies. There is a natural embedding $S \hookrightarrow DS$ whose image is one of the two copies of $S$ in the quotient $DS$.
Next, in order to avoid a very small number of pathological cases, let $\widehat{DS}$ be the connected sum of $DS$ with a closed surface of genus $2$, where the connected sum operation is done by removing the interior of a closed disc disjoint from the subset $S \subset DS$, and hence the embedding $S \hookrightarrow DS$ induces an embedding $S \subset \widehat{DS}$.
The surface $\widehat{DS}$ has empty boundary and is not a 2-sphere, projective plane, torus, Klein bottle, open annulus, or open Möbius band, and hence $\widehat{DS}$ has a complete hyperbolic metric (by an application of the uniformization theorem). By restriction one obtains a hyperbolic metric on $S$.