The Nieuport 13 was a two-seat biplane similar to the Nieuport 12, but originally built with a lower powered engine, possibly for use as a trainer. The aircraft was later modified to use a more powerful engine, and may have received weapons, but didn't enter service in either configuration.
The Nieuport 13 is one of a large number of Nieuport designs about which very little concrete information has survived. Some Nieuport documents discuss a Nieuport XIIIB that was a two-seater with the same wing area as the Nieuport 12, but with an 80hp Le Rhône engine. This was a drop of 30-50hp compared to the Nieuport 12, and suggests that this might have been a trainer (similar to the later Nieuport 80 series). One surviving photo probably shows the Nieuport XIIIB. If this is correct then this aircraft had a different arrangement of central struts to the Nieuport 12, with two struts on each side of the fuselage, angled outwards as they rose to connect to the upper wing.
Other Nieuport documents and a second photograph suggest that the Nieuport 13 was later given a more powerful 150hp Hispano-Suiza engine, and an Etévé gun mounting in the rear cockpit. This was a water cooled inline engine, so required a new engine mounting. The surviving photograph showed a wedge shaped nose, with bulges on top to give space for the cylinders. A similar engine installation was used on the short-lived Nieuport 14.A 2 bomber.
To add to the confusion the Russians used the designation Nieuport 13E for some 80hp Le Rhône powered trainers, although it isn't clear if these aircraft had the modified struts.