Big question! A rule of thumb may be: "would" is related to "will" and "could" is related to "can". So "could" somehow implies "capability".

"Would" is the past form of will and it's often used in similar ways:

Indirect speech:
Tomorrow will be fine.
The forecast said the next day would be fine.

Future in the past:
In 1945, she met the man whom she would one day marry.
Chances are that my letter would arrive in time.

Interpersonal use:
Would you give me a hand, please?
Would you mind waiting for some seconds?
I'd like to speak to John for a moment.

Past habits:
When he was young, he would play chess for hours. (would = used to)

And many more, I guess!

"Could" (like "can") is used to talk about ability, possibility, probability and interpersonal requests:

I couldn't hear a single word! (ability)
You could get a better job if you spoke a foreign language. (conditional)
I aked if you could phone me this evening. (reported speech)
It wa a place where anything could happen. (possibility)
It could rain later this evening. (probability)
Could I ask you something if you don't mind? (permission)
Could you lend me some money? (request)
You could have told me you were not coming. (criticism)

And a huge et cetera!

Hope this helps!

Raul