In a client/server scenario, it may be that only the server has a certificate, so that the clients can confirm the identity of the server; or it may be that clients too are required to have certificates, so that the server can identify their identities.
If only the server has a certificate, the online service is said to be operating in server-only authentication mode. If the clients too have certificates, the service is operating in full authentication mode.
If as a Web site owner you want the benefits of non-repudiation, that is, you want to be able to prove that data really was sent to you from a particular client, then you should use full authentication.