Unspent convictions are convictions that the Rehabilitation of Offenders Act (ROA) 1974 determines not yet to have reached the point where the person is to be treated, for most purposes, as if they had not committed the offence.
Some convictions can never become spent, e.g. any sentence of imprisonment for more than 2 years and 6 months will remain unspent for life.
For England and Wales, a full list of offences that may never be spent is available here.
For Scotland, a full list of offences that may never be spent is available here.
The point at which a conviction becomes spent depends on the sentence received and not the type of offence.