Would has three major functions:
1. It is the past tense counterpart of will.
2. It plays a major role in hypothetical or contrary-to-fact conditionals
in the result clause and at times in the if-clause, as well
as in wish-clauses.
3. Act acts as a “downtoner” or “softener”
in interpersonal (pragmatic) interactions.
As the past tense counterpart of will:
He said they would print a retraction immediately.
(future in the past in reported speech)
We all realized that she wouldn’t get better.
(future in the past in indirect speech)
After a colorless first term, he would go on to
become the nation’s greatest wartime president. (future in
the past in main clause, literary style)
They would arrive just after we had put all the
food away! (describing annoying behavior; compare to: "He will
track mud all over the house!")
Note: would is also used to describe past habitual
behavior (as long as a past context is established) as in:
He was quite a character. He would tell stories about
his adventures on the high seas that made us roar with laughter.
Some examples of its use in hypothetical and contrary-to-fact statements:
If I were Senator X, I would be furious. (the result
clause of a hypothetical or contrary-to-fact present conditional)
If you would (just) give me a few more days I’d
be forever grateful. (an if-clause expressing volition)
As a “downtoner” or “softener” in pragmatic
interactions in the present:
It would appear that he is inordinately interested
in her health.
I would wait a bit longer before going to your lawyer.
Would you like to help me in the
kitchen for a minute? [Also: I’d like to go over these figures
with you (as a “softened” version of want)]
Other pragmatic uses include would rather, would prefer,
and would sooner.
See More for Rachel’s comments and examples