Answer:
Manchester is a city and metropolitan borough of the county of Greater Manchester, which was established on the 1st April 1974, following the Local Government Act of 1972. This metropolitan county also includes the Metropolitan Boroughs of Bolton, Bury, Oldham, Rochdale, Salford, Stockport, Tameside, Trafford and Wigan.
Prior to the 1972 act, Bolton, Bury, Manchester, Oldham, Rochdale, Salford and Wigan were in the county of Lancashire but were administratively independent of the county because they were County Boroughs. Stockport was a County Borough of Cheshire.
Manchester attained city status in 1853 and between 1890 and 1933, incorporated various Cheshire villages (such as Burnage, Didsbury and Withington) to the south of the river Mersey.
The 10 boroughs of Greater Manchester had a county council until they were dissolved by the Thatcher administration in 1986 in the Local Government Act 1985, and many of its functions were transferred to the respective boroughs and to joint boards, such as the Greater Manchester Passenger Transport Executive (GMPTE) [http://www.gmpte.com/]. Voluntary boards such as the Association of Greater Manchester Authorities (AGMA) [http://www.agma.gov.uk/] aim to maintain the inter-borough service integration that was delivered by the metropolitan county council. The county remains a geographic point of reference and a lord lieutenancy [http://www.manchesterlieutenancy.org/].
The confusion around the county in which Manchester lies is said to mainly stem from the fact that the Royal Mail never adopted Greater Manchester as a postal county, citing potential confusion with the postal town of Manchester- postal counties were no longer required on mail from 1996 [ftp://ftp.royalmail.com/Downloads/public/ctf/rm/PAF_Code_of_Practice_March_2004_2.pdf]. In addition to this, it is often assumed that the dissolution of the county council of Greater Manchester equated to the dissolution of the metropolitan county over which it presided. Finally, the fact that Lancashire County Cricket Club play their home games at Old Trafford cricket ground in Trafford, Greater Manchester also leads to the misunderstanding that Manchester is in Lancashire.