A broker-dealer (B-D) is a person authorised to trade securities on its own account (dealer capacity) or on behalf of fiduciaries (broker capacity).

Clearing refers to everything that takes a trade from transaction, e.g. a voice confirmation over the phone, to settlement, i.e. securities and cash having changed hands. Clearing involves such processes as making sure you (and your customers) remain within margin constraints, mailing tax documents, dealing with counter-parties messing up and sending you wrong securities/amounts of cash or forgetting to send you anything at all (a failed trade), etc.

A self-clearing B-D, instead of outsourcing this function to a bank's "prime brokerage" function or a dedicated clearing outsourcing firm, has its own operations department that takes care of these functions.