A clearing house provides clearing services for trades that take place through an exchange. The exchange matches buyers with sellers, the clearing house then has to deal with the transfer of ownership.
The clearing house used by the London Stock Exchange is LCH Clearnet. In the US the Depositary Trust & Clearing Corporation (DTCC) has a monopoly on clearing services.
It is common for a clearing house to guarantee trades by acting as a central counterparty (as happens for most securities trading in the UK). It is also common for a clearing house to provide depositary services, recording ownership of dematerialised securities as DTCC does.