CHESS (Clearing House Electronic Subregister System) is a computer system used by the ASX for shareholding bookkeeping. Holding CHESS-sponsored shares means that those shares are recorded by the ASX directly. When using a broker offering CHESS-sponsorship, you are represented on the ASX’s system by an HID (Holder ID number) they assign you.
Holding shares without CHESS-sponsorship means that someone else is holding those shares on your behalf