A background check by an employer in India usually entails verifying the data and references provided by you. They want to make sure you are being truthful about your past employer, designation and period of work. Some firms also request a "character certificate".
A "character certificate", from a private or government employer, is mainly meant to show that your discharge from an organisation was under normal circumstances. And that you didn't abandon the job without notice or were terminated because of any unethical and illegal action by you. A character certificate is only supposed to be a reference to your moral character. For example, it doesn't (and shouldn't) mention anything about your poor work performance or any disciplinary actions your faced in the organisation, if any.
Most companies issue this as a routine one liner in the termination letter or relieving letter - "[Your name] worked as [designation] from this [start date] to [end date] and his / her conduct and character was satisfactory / excellent / upstanding during his employment here."
So unless you got caught lying or cheating by your previous company in some form, and consequently lost your job, you don't really have to worry about anything.
As I mentioned, background checks are mostly verification. This is why most companies demand a termination letter / relieving letter or a salary slip of your previous employer from you before joining. It makes their job easier of verifying the data you submit. This is what is happening in your case too - your future employer is asking you to get a reference letter from your previous employer. (And that is why it is important to keep in mind to always ask for a relieving / terminating / reference letter from your employer when you leave them).
So ask your former employer for a hard copy of a terminating / relieving letter - tell them companies are asking you to submit it first as part of the process of the background check. And that you would be willing to come and collect it personally. They are legally obliged to give you this. (It is better to swallow your pride and go meet the concerned person in HR directly and request for the hard copy, and then send them an email reminding them of your meeting and request. At worst, if they are unprofessional, they may delay the process and make you come to the office a few times. Bear with it. Worst case resort, you can simply ask a lawyer to make the request for you, and the company will immediately comply).
Also note that many indian companies understand that minor misunderstandings - where smaller companies threaten to withhold termination letters - are quite common.
Tip: Note that a terminating / relieving letter only states the bare minimum facts of your employment and is issued by your employer. A reference letter from a former boss or client however may be more detailed and also help with future employment - always try to get a reference letter too from a senior colleague when you leave a job.