Usually what a private banker would do is to help their clients plan, invest, borrow and bank to create the life and legacy. In simple words, they provide investment advices to help clients preserve and grow their wealth for multiple generations. Clients usually are tycoons, entrepreneurs, industry leaders and their families. Source: PWMA website 

If you are referring to this famous apprenticeship programme organised by Private Wealth Management Association (PWMA), apprentices supposedly can rotate from front to middle and back offices to learn Investment Products, Client Services, Risk Management, Compliance and On-boarding in WM landscape. It is such a great Programme that you can tell from the above participating banks. 

But of course and to be frank, as an intern, you shouldn’t expect you will have heavy job duties say direct deals or client management tasks, but just to prepare market updates for teammates. It is good enough to expose yourself in this field and see if you like it.

Btw, be cautious about this term “wealth management” if you are not applying to that mentioned programme, as so many agencies hire “interns” via this title. Though there is a huge amount of quality internships offered by large banks say UBS, JPM, BNP, HSBC and SCB. Good luck!