An economic system in which goods (e.g. your toys, your food) and services (e.g. your nanny/babysitter) are produced (if material) and exchanged on markets which are largely unregulated (don't have very many rules) in order to make a profit.
The idea is based on that of a society having two classes of people: those who own capital and those who work to produce said goods or provide said services.
Cold wars occur when two superpowers, for example the USA, Russia or China, are aware that a "hot war" (with guns and ships and bombs and whatnot) would cause what is known as MAD, or Mutually Assured Destruction.
The only cold wars in human history have occured between nuclear superpowers, and all of them have been in the last 80 years.
While a hot war has people on the front lines, actually fighting each other, a cold war is primarily espionage (spying) and stealing or sabotaging technology and weaponry.
The Space Race, for example, was a major component of the Cold War between the USA and USSR. Rather than fighting a normal war, the countries instead competed with technology.
The Vietnam War and Korean War are what are known as "Proxy wars", which is when two superpowers in a cold war send supplies and troops to opposing factions of a smaller, regional war. Due to the nearly unlimited military budget of most superpowers, proxy wars are some of the most deadly conflicts in history.
Communism is a moneyless, stateless, & classless envisioning of society. Under communism there is no "private property" as there is today. Instead communists believe in a form of "personal property" that is determined by use of an object (as opposed to abstract ownership).
For instance: a house. If you eat, sleep, and just generally live in a house it is yours. You own it because you use it. That is your personal property. A house becomes private property when you no longer live in it, but still maintain ownership over it and charge others for the right to use it (in other words, renting property).
Communism rejects the concept of private property and it's effects on society, the economy, and production. They support common ownership of the means of production (anything that is used to produce a commodity) and of natural resources.
tl;dr Communism is a stateless, moneyless, and classless. No private property. Common ownership of the means of production.
Exterminating the Jews was a side project for the Nazis, not the focus of the war. What they wanted was land. It was a philosophy called "Lebensraum" which translates to living room. They literally wanted more space for Germans to live in, more land for them to farm, more factories for them to work in, etc. Exterminating the Jews, Gypsies, Homosexuals, 7th Day Adventists, Disabled, and others that they considered sub human was a matter of purifying the German bloodlines. This was the concepts of Eugenics gone mad.