The first paragraph of ABerrant's post is correct with regard to 'speeding' points (although, as his/her link will show, points can remain on a licence for up to 11 years for more serious motoring offences).
However the second paragraph is incorrect. The Rehabilitation of Offenders Act means that offences resulting in a fine become 'spent' after 5 years. If an insurer or employer ask you if you've ever been convicted of a driving offence, after 5 years have passed, you can lawfully answer 'No'. The Act also relates to matters 'ancillary to a conviction', such as penalty points on a driving licence.
Chris