An accepting house is a financial institution that accepts a bill of exchange. A bill of exchange is a financial instrument closely related to a cheque - in fact, a cheque is a bill of exchange drawn on a bank.
When an accepting house "accepts" a bill of exchange, it guarantees it. The bill can then be more easily sold, as this greatly reduces the credit risk.
From most investors' point of view, accepting houses are not a hugely important part of the financial landscape.