"Private" (rank) is often translated to "Gefreiter" which is not the correct translation.
The "Private" is equivalent to the lowest ranks e.g. "Schütze" and "Kanonier" in the army, "der Flieger" in the air force and "Matrose" in the navy.
"Gefreiter" is equivalent to the "private first class" (PFC) in the US. and the "lance corporal" in the UK.
In the NATO armed forces, the translation problem is more complex. The US Army and Canadian Forces each know two private ranks, which distinguish between the untrained recruit (USA: Private E-1; Canada: Private Recruit) and the graduate of basic military training (USA: Private E-2; Canada: Private Basic). The Private E-2 and Private Basic are very close to (Swiss, Austrian and) German "Gefreiter". In the Bundeswehr "der Gefreite" has more rights and duties, though. In the NATO they are - because of their similarity - treated as "Privates" and also receive the "Private" OR-2 grade.