As far as the United Kingdom is concerned, a 'county borough' was an urban area which was a county in its own right. (Roughly equivalent to the status to an Independent City in the United States.) Hence although a 'county borough' might geographically be considered as part of a particular county, it was an entirely separate administrative unit for the purposes of local government. So that Coventry for example, might be in Warwickshire but was not administered by Warwickshire County Council but by its own county borough council.
They were created by the Local Government Act 1888 which established a total of sixty-one county borough councils in England and Wales(amongst other things), which replaced the previous medieval counties corporate and municipal corporations that had been converted into local authorities by the Municipal Corporations Act 1834.
There were subsequent additions to that number as well as later amalgamations, boundary changes etc and an extension of the concept to Scotland and Ireland. (Later contracted to just Northern Ireland.)
All county boroughs were abolished by the Local Government Act 1972 which took effect in Northern Ireland in 1973, in England and Wales in 1974 and in Scotland in 1975. (See below for the best available list of county boroughs as existed just prior to their abolition.)
Just to complicate matters the Local Government (Wales) Act 1994 then re-introduced them by allowing Welsh county councils to call themselves 'County Borough Councils' or 'Cyngorau Bwrdeistref Sirol' (to give the Welsh equivalent. There are eleven councils in Wales that have taken this option, mostly the smaller urban areas or towns; the larger Swansea and Cardiff, seemingly happier to remain as simple 'county'.
Hence all the county boroughs that currently exist are in Wales.
The historic County Boroughs of the United Kingdom
As sourced from the GENUKI website
Of the more recent changes; Middlesbrough was a county borough until 1968 when it became part of the expanded Teeside, and West Hartlepool was a separate county borough until 1967 when it merged with Hartlepool. I would imagine that there were many other changes during the previous eighty odd years.
Welsh County Boroughs created by the Local Government (Wales) Act 1994
As per Schedule 1 Part II of the afore mentioned legislation, with the Welsh equivalent in brackets where applicable.