do I need an invitation letter if I want to visit my sister who lives in China at the moment.

Hi Susanne! If you will stay with your sister's family rather than stay in a hotel during your China trip, you should get an invitation letter from your sister's husband, and perhaps a photocopy of his passport and residence permit in PRC. By doing so, you don't need to show round-trip tickets and hotel bookings when applying for the tourist L type entry permit. So you don't need to buy the air tickets first.