The Administration was a body involved in the running of the Federation, although exact details of the connection between the two were never explicitly revealed.
Citizens of Earth usually referred to "the Administration" rather than the Federation when talking about the authorities, so it may be that the Federation controlled the colonies while the Administration controlled Earth.
However, the two were clearly closely linked, with the Administration able to set policy for Federation armed forces (it was the Administration who ordered the Liberator be captured intact) and also held responsible for the security of the colonies (at one point, Servalan is informed that Outer Planet authorities are displeased with the Administration's defence system).
It seems most probable that the Administration was simply the governing body and civil service of the Federation.