They are called "puppets". This term is somewhat derogatory and is usually used only when the leader is not supposed to be a puppet.
Incidentally, a Governor-General in a state in the British Commonwealth, a bit like an ambassador, is indeed a "puppet" although it would be derogatory to say it in those terms because that is the intended nature of the relationship. Now, however, the puppet relationship is largely de facto between the country's prime minister and the Governor-General, rather than between the British sovereign and the Governor-General as this post is de jure.
A ruling coalition that is made up of puppets is also sometimes called a "Vichy" regime after the regime in France during World War II that was made up of puppets of Nazi Germany.
The term "quisling" is also used for a "puppet" with the stronger connotation that being a puppet in that case is definitely affirmatively traitorous to the country ruled by by that person. The term quisling can also be someone who isn't powerful enough to be considered a puppet, such as a lower level functionary of a traitorous Vichy regime. (Per Wikipedia: "The word originates from the surname of the Norwegian war-time leader Vidkun Quisling, who headed a domestic Nazi collaborationist regime during World War II.")
Both "Vichy" and "quisling" are strongly derogatory terms.
The term "proxy" is also sometimes used for a leader or government who advances the cause of another state (usually a more powerful one or coalition of more powerful ones) at least in some particular area.
For example, North Vietnam's leaders were seen as proxies for China, and South Vietnam's leaders were seen a proxies for the United States, in the Vietnam War (as it is called in the U.S.). This war is often described as one of several "proxy wars" that occurred during the Cold War period.
A "proxy" is not necessarily a "puppet" in all aspects of governance. The term may be issue or policy area specific.
The most common term for someone like Paul Bremer who overtly rules a territory in the name of another government or coalition as a colony, dependency, or conquered territory with no pretense of democratically representing the governed territory at all, is usually a "governor" (even though that term can also refer to the democratically elected leader of a sub-national unit of government such as a U.S. state). Someone who governs some or all governmental functions in a regional territory within a larger government who is appointed by and responsible to to a higher level of government, rather than the people of that territory, is sometimes called a governor, but is also sometimes called a "prefect", "receiver", or "trustee".
Client States Distinguished
A less extreme case than a puppet state is a client state, which refers to a government which is deeply reliant upon another state for its existence or well being.
For example, Israel and Taiwan have been, at least at times, client states of the United States, whose existence would be forfeit or deeply imperiled without U.S. backing, due to strong threats from other powerful countries.
Similarly, Cuba was a client state of the U.S.S.R. for many decades.
But, a client state can have far more autonomy than a puppet state, at least if the sponsor state permits it.
When there is a puppet state, the implied beneficiary of the relationship is the state which "holds the strings" of the puppet. In a client state, the client state is benefiting from its relationship with the sponsor state, even if that means that some circumspection and deference by the client state to the sponsor state is necessary as a result.
