In the Old Days, one country would conquer another and add it to its empire. What was the most recent country to be conquered and cease to exist as a country?
One candidate for "last widely recognized country to be conquered" would be South Vietnam. From the Wikipedia article:
Despite a peace treaty concluded in January 1973, fighting continued until the North Vietnamese army overran Saigon on 30 April 1975.
Its sovereignty was recognized by the United States and by some eighty-seven other nations. It had membership in several special committees of the United Nations, and would have been a member of the United Nations itself had it not been for a Soviet veto in 1957
I started my search by looking at Wikipedia's list of former United Nations members. Of these, some are countries that ceased to be members because they were absorbed by other countries:
- East Germany (became part of West Germany; peaceful transition)
- Tanganyika and Zanzibar (apparently peaceful)
- Formation of Malaysia (apparently peaceful, or in any case formed from colonies that were not independent until absorbed)
- North and South Yemen (apparently peaceful)
A reference to North/South Vietnam in the Yemen article got me to think about South Vietnam, which is perhaps an obvious answer to the question. There might be other, non-recognized states: I can think of Biafra, independent from 1967 to 1970 (so in any case earlier than South Vietnam).
In terms of "empire" conquering that has had significant cultural effect, meaning a large change in the ethnic or linguistic nature of a large area of land the biggest changes in the last 100 years:
- Elimination of Prussia (1945-1948)
- Elimination of Manchuria (1930s)
- Collapse of the French empire in West Africa
- Collapse of the British empire in Africa, India and Southeast Asia
- Creation of Pakistan, Burma and Thailand
- Elimination of Austro-Hungarian Empire and creation of Czechoslovakia, Croatia, Slovenia and Hungary
- Collapse of Ottoman empire, creation of Israel, Iraq, Syria and Kuwait
The most recent significant area to be "conquered" and absorbed by another country is the Crimean peninsula, conquered and annexed by the Russia Federation this year. The most recent time an entire "country" has been conquered and annexed militarily by another country was the de facto annexation of Republic of Chechnya in 2009.
Arguably, Western Sahara could also fall in this definition. It was decolonized from Spanish rule in 1975 and soon after been occupied by Morocco and Mauritania. Later on Mauritania retreated and Morocco took over most of the land leaving a small territory in the hands of the Polisario Front, a western Sahara's people guerilla movement.
So, this is not a clear cut case as Western Sahara wasn't a sovereign state and it has not completely vanished but it's recent and fairly close to what you're asking. Maybe even more than South Vietnam.
Since some people did not like my last answer I will reformulate it. The German Democratic Republic was a member of the United Nations and was recognised by most countries in the world, including the USA. In 1990 it disappeared from the map and was incorporated into the Federal Republic of Germany. It was not conquered militarily by West Germany, but it was annexed by it.