If the ownership of a security is recorded in a register kept by the issuer (or a third party registrar) it is a registered security. If its ownership is not recorded it is a bearer security.
Shares are almost always registered securities but ADRs and eurobonds are usually bearer securities.
If a register exists, it is then usually the main evidence of ownership of the securities. It is more important than paper certificates. Certificates and other written evidence of ownership may be useful for trading, when tracing fraud or for tax returns.
It is now usual for shares in listed companies to exist in dematerialised form. No certificates are issued and the shares exist only as ownership registered in depositary systems and registers.
It is also common for securities to be held by stockbrokers' nominee companies. The broker's clients who bought the securities are the beneficial owners. There are problems with this arrangement as these shareholders do not always receive the same rights (particularly to vote at AGMs and EGMs) as those whose names appear on the register.
Maintaining the register of shareholders is the responsibility of the company secretary. Most companies of any size find it more cost effective to outsource most of the actual work to a specialist registrar.