The equivalent organization in the United Kingdom to the FBI (Federal Bureau of Investigation) is known as MI5.
This is in fact a misnomer: the original office for the section was in Room 15, on the mezzanine floor of a government building ... hence M15. This was widely mis-read, and "MI5" has passed into the language. In fact, the 'head office' is no longer in that room.
Like the FBI, its mandate is investigation within the country. The equivalent of the United States' CIA (Central Intelligence Agency) is known as MI6.
The CIA equivalent in the UK is known as the SIS (Secret Intelligence Service or MI6) The MI5 is still the main Secret Service (SS) The FBI and CIA have field agents in the their UK headquarters that operate in The European sector. MI5 and SIS have the equivalent overseas. A large number of FBI/MI5 field agents move from country to country protecting the interest of their homeland securities.
Additionally within the UK the serious organised crime agency SOCA which has been dubbed the British FBI operate but due to the nature of the UK police force and its structure no direct comparison can be directly drawn between UK and US forces