> C to the same C is "unison" (You may need to instruments to play two identical C notes at the same time.)
> C to C#/Db (this note has two names) is "small second"
While C# and Db are the same note (in equal temperament [1]), the intervals C/C# and C/Db have different names: C# is called 'augmented unison' [2]. For the name, you start from the basic interval (e.g. C/C) and apply the accidentals (# or b) [3].
This is an example of why the traditional approach to music theory can be cryptic for a beginner. After the Western music moved to well and equally tempered scales (starting from the early 1700's), the context in which there is a difference between C# and Db has disappeared. But we still use terminology and notation from 500-800 years ago.