Ok. Basically, "would" forms something called the conditional tense. A clause in the conditional tense say what would happen if something else happened. Here are some examples:
If I were happier, I would jump with joy.
Here's a diagram of what this sentence means, in logical terms:
Happier -> Jump with joy
The use of the past subjunctive "I were" (don't worry about that, it's complicated)
happier is not true. (I am not happier.) So does "would jump with joy," which implies that I'm not jumping with joy right now. But, provided that
happier were true,
jump with joy would be true.
Note that this is very hard to explain without using if or would. Even provided that is really just another way to say if.
Here are some more examples, with their meanings:
If I ate more food, I would become fat.
Eat food -> become fat
If you were nicer, we could be friends.
You be nicer -> we are friends
I would be angry if I weren't so tired!
Not tired -> angry
Bottom line: the conditional tense is used with an if clause to denote something that probably isn't true, but would be true if something else were true.
EDIT - Here are the meanings for your example sentence:
Presumption or probable past state: They were in the same class, so they would have known each other.
Here you're using would to say something that was probably true based upon evidence, but you don't know for sure.
Necessity: He grew so angry, it seemed he would explode.
Here, you're using would to mean something that isn't actually true, but seems likely.
Uncertainty: So it would seem. So I would think.
Again, this is not necessarily true, but we are saying that it should be based upon evidence.