A “puppet state” is a country that claims to be independent, but is controlled by an outside state or other power. Puppet states are not recognized by international law.
A puppet state has the appearance of being independent. It typically has things like its own flag, constitution, system of government, etc., but still cedes control to another power.
Examples of puppet states include countries that were under the Soviet bloc, like Czechoslovakia, Poland, and Mongolia. Some puppet governments are implemented through military force, like Vichy France in WWII. Many colonized countries in Africa and Asia would become puppet states controlled by their colonizers. Examples of this are pre-1960s Vietnam, India, and the Congo.