James Gregory designed the Gregorian reflecting telescope in 1663, which uses a concave primary mirror and a smaller, convex secondary mirror to focus light. It was the first reflecting telescope to be designed, though it wasn't until 1673 that Robert Hooke built the first working model of a Gregorian telescope.
Refracting telescopes use lenses to focus light, and were invented earlier than reflecting telescopes. The first refracting telescope is credited to Hans Lippershey in the Netherlands around 1608.