  In this paper we introduce the Terzina telescope as a part of the NUSES space
mission. This telescope aims to detect Ultra High Energy Cosmic Rays (UHECRs)
through the Cherenkov light emission from the extensive air showers (EAS) that
they create in the Earth's atmosphere. The Cherenkov photons are aligned along
the shower axis inside about $\sim 0.2-1^{\circ}$, so that they become
detectable by Terzina when it points towards the Earth's limb. A
sun-synchronous orbit will allow the telescope to observe only the night side
of the Earth's atmosphere. In this contribution, we focus on the description of
the telescope detection goals, geometry, optical design and its photon
detection camera composed of Silicon Photo-Multipliers (SiPMs). Moreover, we
describe the full Monte Carlo simulation chain developed to estimate Terzina's
performance for UHECR detection. The estimate of the radiation damage and light
background rates, the readout electronics and trigger logic are briefly
described. Terzina will be able to study the potential for future physics
missions devoted to UHECR detection and to UHE neutrino astronomy. It is a
pathfinder for missions like POEMMA or future constellations of similar
satellites to NUSES.
