what is a demat account?

Demat is a short name for dematerialised account. Just as you have to open an account with a bank to save your money , make cheque payments etc, likewise you need to open to a demat account where actual money is replaced by shares . So it is just like a bank account where actual money is replaced by shares. You have to approach the DPs (remember, they are like bank branches), to open your demat account. Let’s say your portfolio of shares looks like this: 150 of Infosys, 50 of Wipro, 200 of HLL and 100 of ACC. All these will show in your demat account. So you don’t have to possess any physical certificates showing that you own these shares. They are all held electronically in your account. As you buy and sell the shares, they are adjusted in your account. Just like a bank passbook or statement, the DP will provide you with periodic statements of holdings and transactions.

Though the company is under obligation to offer the securities in both physical and demat mode, therefore you get the advantage to choose either of the mode.
If you wish to have securities in demat mode, you need to indicate the name of the depository and also of the depository participant with whom you have depository account in your application.

It is, however desirable and advised that you hold securities in demat form because sometimes  physical securities carry the risk of being fake, forged or stolen.

Is a demat account a must?
Nowadays, practically all trades have to be settled in dematerialised form. Although the market regulator, the Securities and Exchange Board of India (SEBI), has allowed trades of upto 500 shares to be settled in physical form, nobody wants physical shares any more. It adds to surety and security of the shares and money invested also.
So a demat account is a must for trading and investing.
A broker is separate from a DP. A broker is a member of the stock exchange, who buys and sells shares on his behalf and on behalf of his clients.
A DP will just give you an account to hold those shares.
You do not have to take the same DP that your broker takes. You can choose your own.