Generally the County Council has responsibility for things like transport, education, social services, museums etc. The District or Borough Council covers areas like housing, refuse collection, leisure and so on.
Unitary councils generally cover smaller areas than counties but are responsible for all local services within their boundaries. Many metropolitan areas have unitary councils as do London Boroughs. In the last few years local government has been reorganised so there are more unitary authorities. All of Scotland and Wales now have the Unitary system.
The result of all this was that although Leicester City had held it's four-yearly election in 1995, there had to be fresh elections for the new Leicester unitary authority in 1996. The elections were contested over the same boundaries to elect the same number of councillors and so can be freely compared to each other.
Rutland District council also went through a similar procedure. In 1995 the normal four-yearly full district council elections were held. However in 1996 elections were held using the same ward boundaries for the new unitary authority of Rutland.
Leicester City and Rutland both legally became unitary authorities in April 1997. The elections were held a year early so that the new councils could "shadow" their predecessors and plan for the extra responsibilities of unitary status.