The term “onshore banking” (onshore private banking or onshore wealth management) refers to the practice of managing wealth in the asset-holder’s country of residence. In this way, it differs from offshore banking in which assets are managed outside of their owner's country of residence.
For example, if a customer residing in Switzerland has their assets managed in Switzerland rather than in another country (like Singapore or the U.S.), they are using onshore banking services.
More on this topic:
Offshore banking explained
Private banking in Switzerland