Why so many bitcoin wallets do exist? Why it is not just single like other conventional currency? Multiple wallets means I have to have multipe accounts; is it possible to transact among multipe wallets bitcoins and if yes is it without fee? For bitcoin account, public key is actually address which will be used in bitcoin transaction (sending bitcoins to me). And private key is used to decrypt the transaction. I also got to know that the key can also be changed. If my understanding is correct, what is the actual key for the bitcoin account?
What exactly was a bitcoin wallet intended to be?
Multiple wallets means I have to have [multiple] accounts;
The original creator of Bitcoin, Satoshi Nakamoto, wanted to create a currency for internet use which did not depend on any trusted third parties! Originally all wallets were software programs that you downloaded and ran on a personal computer. You would only have one wallet, not multiple.
There were no accounts, no exchanges or other businesses that you had to trust with your money. That was deliberate. You can still use Bitcoin like this today.
Over time people and businesses have seen an opportunity to offer services to people who either don't understand the original intent or who want to treat the currency not as a currency but as a speculative asset to be traded. These businesses have set up custodial accounts and have often described those accounts as wallets. These "custodial" wallets are not the type of wallets that the founder of Bitcoin intended.
You don't have to have multiple accounts. You don't have to have any accounts. You can use Bitcoin with no accounts whatsoever. Just download the wallet software of your choice, transfer all your Bitcoin to that wallet and close all your other accounts.
What is electronic cash?
Consider the way most USA citizens use US dollars. Very few of them ever buy and sell US dollars. They just receive their income in US dollars and spend their money at businesses that accept US dollars for goods and services. There is never any need for these people to buy US dollars or to "cash out" their US dollars.
This was how Satoshi Nakamoto and other early adopters hoped that Bitcoin would be used.
Two hundred years ago, many US citizens would have received their wages as US dollar bills (banknotes) and would have used those for all their spending. They wouldn't need a bank account at all. Their finances could be totally cash based.
Nowadays many countries are moving rapidly towards a cashless society where all money is controlled by banks.
Nakamoto wanted a return to a cash based system where people, not banks, controlled their own money. But he wanted it based on an electronic currency that could be used online and which wasn't controlled by governments - because governments use inflation to effectively take money from savers.
So Bitcoin was intended as a return to a cash-based society, but one that could be used online and not subject to government or bank control
So each person would have a single wallet, no accounts.
You can still use Bitcoin like this.
The role of Cryptography
private key is used to decrypt the transaction.
The Bitcoin network does not use any encryption. So no decryption is necessary.
The private keys are used to provide a mathematical proof that you have the right to spend amounts of Bitcoin associated with a Bitcoin address (or more correctly, associated with a Bitcoin script that can usually be represented as a Bitcoin address).
Private keys are effectively used to prove ownership of money.
I also got to know that the key can also be changed.
Not really, however you can use a transaction to move money to the control of a different private key.
what is the actual key for the bitcoin account
The Bitcoin network doesn't know about or keep track of accounts. Many Bitcoin wallets don't have any sort of account concept.
If you have an account with some sort of trusted third party, the way that works is proprietary and may be different for each business.