A gilt edged market maker is a dealer authorised by the Bank of England to make a market in government stocks (gilts).
The firm establishes the price of stocks in the same way that a market maker in shares does. It's required to offer 2-way prices and make a continuous market in gilts.
In return for this obligation it has direct access to the Bank of England and may have borrowing facilities at the Bank. The firm may also receive favourable tax treatment.