Can I Apply For My British Passport While I Am On My Ancestral Visa Living In Britain?

The ancestral visa is granted to the first generation of grandchildren, of British Born people. It only applies to the commonwealth countries, so for example, you came in to work and live, on the strength of your grandparents nationality. Generally you must stay five years continuously and not be out of the country for more than 90 days a year. Then after the five years is up you can apply, (providing you have a job, can support yourself, and haven't done anything against the government ) to become a naturalised citizen. after which time you could obtain British Passport as you will be given British Nationality.
If you have more questions, seek advice from the citizens advice bureau, or an immigration lawyer.