The interest rate is the cost of borrowing money, expressed as a percentage. When you borrow, you pay interest on top of the principal; when you save, you receive interest on your balance.
The Bank of England sets the base rate — the interest rate at which it lends to commercial banks. Commercial banks then set their own savings and mortgage rates based on (typically above) the base rate. When the base rate rises, mortgage rates and savings rates generally rise; when it falls, both tend to fall.