What is the difference between "I would be really grateful" and "I will be really grateful"? In particular, which one should be followed with "could"? I mean like this "I would be really grateful if you could answer" or "I will be really grateful if you could answer": which one should I use?
A previous question asked 'Which dialects of English consider “would” to be a polite form of “will”?' I'd say it's a virtually universal pragmatic device to use 'would' where 'will' would appear the logical choice (though there are probably some situations where using it might be confusing).
Here, 'I will ...' might sound too presumptuous for some, getting too close to assuming compliance. But I'd say that it's a matter of pragmatics rather than grammar (ie don't worry about breaking 'rules' that might have been suggested in some 50-year-old book).
Where there is no sense in hedging, 'will' is quite acceptable, but doesn't pair with 'could':
'I will be really grateful if Limeside Council answers.'
Rob De Decker, at Grammar.ccc.comm, agrees that politeness-marking can trump expected usage here, though he says that 'will' can also function as a politeness marker [where volition rather than mere future possibility is obviously implied]:
The conditional construction does not normally use will or would in if-clauses.
EXCEPTION: If will or would express willingness [/ mark politeness], as in requests, they can be used in if-clauses.
e.g. If you will come this way, the manager will see you now.
I would [/should] be grateful if you would [/could] give me a little help.
I would is usually more polite. It gives more of the idea I understand if you cannot help me.
I will sometimes gives an idea like You must help me or I expect you to help me.
We more often use very instead of really, but really is ok. It is also ok to just say grateful without really or very, because grateful is already a strong word by itself.
You can use would or could after you (see below). If you do not use would or could, then you use the past participle (past form of the verb), like helped in the last example below.
Here are some correct expressions:
I would be [very] grateful if you could help me practice English.
I would be [very] grateful if you would help me carry this piano upstairs.
I would be [very] grateful if you helped me clean my room.