'On Saturday' refers to the nearest future Saturday only. 'On Saturdays' refers to each Saturday, either past present or future. So for example, "I teach on Saturdays" means I teach every Saturday. If on Wednesday I say to you, 'I'll see you for lunch on Saturday" I mean the upcoming Saturday only.

On Saturday - most probably the upcoming week (or a Saturday that has been mentioned previously in the conversation). On Saturdays - every Saturday