The Anti-Federalists feared that if the Constitution was ratified (became law) that it would give far too much power to the federal government over the states--leading to tyranny and the trampling of individual rights.
NATO or the North Atlantic Treaty Organization is a military alliance formed in 1949 as a mutual defense treaty of its member countries against external attacks. It was initially formed to counter the military capabilities of the USSR. It currently has 28 member countries from North America and Europe.
The Warsaw Pact was the Soviet Union's response to NATO, and was formed in 1955. It is also a mutual defense agreement among 8 member countries in the Soviet Union and Central and Eastern Europe.
the Democratic Republicans
Democratic Republicans believed in state power over the federal government and the ability of states to ignore unjust laws. In the case of the Whiskey Rebellion, farmers rebelled against a tax on whiskey. They believed it was the right as citizens to refuse to pay a federal tax because they didn't believe in the tax and weren't fairly represented. The rebellion was met by federal troops and a statement of federal supremacy.
Democratic-Republicans formed under the leadership of Thomas Jefferson. The party had roots in the ideas of Anti-Federalists during the Constitutional Convention. The Democratic-Republicans believed the Constitution should be followed strictly and as stated. They also supported states rights over the federal government. In addition they opposed the National Bank and any institutions that gave the federal government power.
Search and seizure would mean something like the government taking your cell phone and going through every message you've ever sent, or entering your home and searching through all your dresser drawers. They need to have a court-approved reasonable cause for doing something that invasive.
Quartering of soldiers would mean soldiers would have the right to enter your home and expect you to provide them food and lodging. That's pretty invasive too.
In either case, we're talking about invasions of your privacy, your personal space. Court justices have used statements like those about search and seizure and quartering of soldiers to show that the constitution does give attention to citizen's right to privacy, even if not using the exact term "right to privacy."