Firms who are not direct members of a clearing house must rely on having their trades cleared by an intermediary, who is a direct member of the Clearing House.
A business model used globally with Clearing Houses is a two tier membership structure, with the first tier being direct Members, who contractually face the Clearing House and must respond to all requirements and liabilities as a result of their membership. In the US these firms as known as Futures Commission Merchants (FCMs) although nowadays they clear a wide range of products, not just futures. Outside the US these firms are generally known as Clearing Brokers (CBs).
Firms who choose not to become a Member of a Central Counter Party (CCP) are generally known as Clients, hence the term Client Clearing.
Economically and contractually a new trade for a Client (with their original Counterparty) is presented to the CCP by a Member, and assuming it is cleared is then split into this contractual structure (outside the US):
Outside US: Client Member CCP Original Counterparty
Within the US, the FCM model has the Client in a direct relationship with the CCP legally, but with the FCM acting as a guarantor of sorts for default management and settlement:
US: Client CCP Original Counterparty (but with settlement being via the FCM)
Any firm acting as an FCM or CB must settle all liabilities required by a CCP including Variation Margin, Initial Margin, Default / Guarantee Fund contribution, plus any capital charges on their own business for holding the Client Trades.