A prospective employer is interested to buy out for early release, as they need to fill up the position urgently, can't wait for notice period to lapse. How does it work?

How to be sure that there's something solid from the prospective employer, in order to bring up the issue of resignation to current employer? Will there be an offer letter? Is it possible to get an offer letter when the starting date is yet to be firm?

And what if current employer refused to accept the buy out and disallow an early release, can it happen?

Would like to know how does this work, thanks guys! 

THIS!

It's not so complicated.

You resign. You then either serve the notice period (however long that is) OR you pay the specified amount in lieu of notice. The next employer simply helps you pay in lieu of notice so that you don't have to serve the notice period and can start with them as soon as you hand in your resignation.

Your current employer cannot refuse to allow you or anyone else to pay in lieu of notice... unless you were foolish enough to sign a contract saying that this is not available.

Hey Ace. First things first. A buy out =/= early release. They are 2 different things:-

1) A buy out is payment-in-lieu of notice by your new employer to your current/past employer.
2) Early release is the act of waiving compensation (i.e. payment-in-lieu of notice) by your current/past employer for you leaving your current employment early
Now to answer your questions:-

A prospective employer is interested to buy out for early release, as they need to fill up the position urgently, can't wait for notice period to lapse. How does it work?
Its usually a 2 stage process.

Stage 1
1) The prospective employer informs you that they will buy out your notice period.
2) Prospective employer issues you a letter of offer, incorporating the above.
3) You accept (execute) the offer letter.
4) You issue a letter of resignation to your current employer. In your resignation letter, you (politely) request for an "Early Release". Write down that you will assist and fully cooperate with a proper and smooth handover. This helps.
5) In your letter, DO NOT put a date for your last day of work as your current employer may take it negatively (you might be seen as rude - dictating the date of your last day at work) (based on my own experience and from my buddies in HR).
6) Allow your current management about a week or so to digest your notice of resignation and to come back to you with an answer to your early release request.

Now 2 things might happen here:-

Scenario 1
Your current employer accedes to your early release request, indicating they will be willing to let you leave early by xx month(s) / xx date (perhaps after adjusting against your remaining annual leave etc). If the early release date is within your expectations, thank your management and inform your prospective employer that you will be joining them on xx date. 

If the date is not within your expectations (for example, you needed a 2 month early release but your employer is only willing to allow a 1 month early release, you can either 1) try to negotiate a date in line with your prospective employer's expectations or 2) accept the early release period and pay-in-lieu the difference.

For example, say you have a 2 month notice period and your current employer is only willing to allow an early release of 1 month,  you then inform your new employer that they will have to compensate 1 month's equivalent notice (2mth notice period - 1 month early release = 1 month pay-in-lieu).

Do note that some employers may not accept payment in-lieu of notice if they have allowed some sort of partial early release. They may just withdraw their early release approval and force you to compensate the full 2 months notice if you attempt the leave immediately. So be clear about their expectations and negotiate fairly and politely.

Scenario 2
Your current employer responds by saying that they will not be able to allow an early release in which case you just need inform your prospective employer that they will have to buy out your entire notice period if they want you in early.
Stage 2
Now that your current employer has responded to your request for an early release (perhaps verbally or via email), follow up with another letter, this time confirming your last day of work and informing your current employer that your new employer will be compensating them in lieu of remaining notice (if applicable).

The above may not be necessary if your current employer responds in writing (there may be some verbal/email negotiation between you and your management beforehand) and confirms your last date of work (either with or without some sort of early release). You can proceed to pass a copy of this letter to your new employer to allow them to process the buy out.
How to be sure that there's something solid from the prospective employer, in order to bring up the issue of resignation to current employer? Will there be an offer letter? Is it possible to get an offer letter when the starting date is yet to be firm?
Yes. You will need an offer letter. Never ever resign without 1) signing the new offer letter beforehand and 2) passing your pre-employment medical examination with the new employer. The starting date need not be a hard date. If can be something like "at the earliest opportunity, which shall be no later than the last working day of November 2012".
And what if current employer refused to accept the buy out and disallow an early release, can it happen?
An employment is a contract. Hence, the employer cannot refuse a notice of resignation (otherwise it will be slavery, not employment). Similarly, an employer cannot refuse a buy out as long as there is "payment in-lieu of notice" clause in your employment contract (which is standard by the way).

FYI, not acceding to an early release request is the norm rather than the exception. But it doesn't hurt to try as you may be able to save money for your new employer.