Bitte um Korrektur! :)

The Cold War

The cold war was a non military conflict between the two "super powers": The USA and the USSR after the World War Two. It was from 1945 to 1990, the reason of the end was the dissolution of the USSR. Both of the countries wanted to be the most powerful country in producing ato, weapons. It was an armament race (Wettrüsten). At the Beginning of the 2. World War they were partners but after the end of the 2. World War they became enemies because of the fact, that the USA was kapitalism (kapitalistisch) and the USSR was kommunistique(keine Ahnung ob das so geschrieben wird ^^) .
They didn't used the atom bomb because they wanted to protect themself and their counry. Thex supported (anstiften?) little countries (Afghanistan, Corea and Vietnam) to have wars, for example, the Vietnamwar.

Vielen Dank ihr hilft mir sehr weiter!!

The Cold War

The cold war was a non military conflict between the two "super powers": The USA and the USSR after the World War Two.

1)"cold war" is a specified term, consequently it has to be capitalized
2) I took me 5 minutes to understand "the USA and the USSR after the World War Two"
It isn't a proper sentence, is it?

It was from 1945 to 1990,

2) "It was" - what is it? Relation?!
3) replace "was" by a "proper" verb like "last"

the reason of the end was the dissolution of the USSR.

4) SPO

Both of the countries wanted to be the most powerful country in producing ato, weapons.

5) No "of" after both. (And not the)
"of" is coming after both for example by "both of them"
6) "wanted to be the most powerful countries. End of the sentence" or you have to rearrange your sentence.
7) ato?

It was an armament race (Wettrüsten).

9) arms race
10) it?

At the Beginning of the 2. World War they were partners but after the end of the 2. World War they became enemies because of the fact, that the USA was kapitalism (kapitalistisch) and the USSR was kommunistique(keine Ahnung ob das so geschrieben wird ^^) .

11) Beginnig - B in small letters
12) Who is they?
13) "were" wrong tense. An action in the past happend before a different action (also in the past).
14) kapitalism= capitalism and capitalism is a noun.
15) kommunistique= communistic

They didn't used the atom bomb because they wanted to protect themself and their counry.

16) Who is "they"?
17) "the" atom bomb? Are you sure?
18) They are not using a atom bomb to protect themselves?
How could they protect themselves by not using the atom bomb?
Did they hide?
19) Which country?

Thex supported (anstiften?) little countries (Afghanistan, Corea and Vietnam) to have wars, for example, the Vietnamwar.

20) support = unterstuetzen
21) small countries
22) Can you "have" a war?
No you can't but you could begin or declare a war
23) War with whom?

Honestly, I do not understand what you are trying to tell us.
I marked most of your mistakes (content), but before we can correct your grammar mistakes you have to think about "the meaning" of every single sentence.

Have fun!