I have seen saddle fixed point/ equilibrium, but I am trying to understand the meaning of Saddle periodic orbit as I encounter them in many research articles like a google search gives me - saddle periodic orbits.
I am interested to know the meaning and significance of a saddle periodic orbit.
For discrete-time systems $x_{n+1}=f(x_n)$, an $N$-periodic orbit is called a saddle if the corresponding fixpoint of $f^N$ is a saddle point (Scholarpedia).
But the paper you referred to deals with continuous-time systems (vector fields). They don't really give a precise definition, but from section 3.2 it seems like what they mean is a just a periodic orbit such that some nearby orbits are attracted (those on the stable manifold of that orbit) and others are repelled. Maybe an easy way to state it more precisely would be to say that it's a periodic orbit such that the corresponding fixpoint for the Poincaré map is a saddle point?