It's not straightforward...

You can ride bridleways but not footpaths, but the O.S. map is not a definitive map which means that even if it shows the prescence or absense of a right of way, it's not legally binding... however 99% of the time if you stick to bridleways shown on the O.S. map you should be okay.

You also have a right of way on normal highways, and certain other byways - 'green lanes', county constructed roads etc. see:

http://www.highwayman.demon.co.uk/bandbt.html

Forest tracks not clearly marked by the landowner as a permitted right of way for bikes should be avoided - note: in some areas the Forestry Commision permit cycling.

Your local council should have a rights of way officer who'll be able to give specific advise.