I believe you are asking about government and politics. In American usage, the term administration refers to the executive branch under a specific president, for example: the "Barack Obama administration.” In Britain, the word "government" is used instead, for example, "the Tory government under David Cameron".

Many constitutional democracies (such as the U.S., U.K., etc.) have a tripartite structure of government consisting of the executive branch (president or prime minister with a cabinet of ministers and a civil service), the legislative branch (e.g. U.S. Congress or British Parliament) and a judicial branch (judges).. Thus, the executive branch is just one part of the government. "The administration" usually refers to the politically elected and appointed members of the executive branch. The civil service refers to the career government workers in the executive branch who are generally not replaced when the administration changes due to political victories and defeats.

They are synonymous when used in a sentence like: "This administration/government has really messed up the country!"

"Administration" is management-- usually some kind of official, legal, or commercial management. "Government" is the management of a political body. There can be a city government, state government, national government, but not company government. (Both can be either an act or a group of people-- management can be either an act of management or a group of people who manage something, for example.)