Q:

What do I need to start trading in the NSE (National Stock Exchange)?

Can I start a Demat account with a bank like the Oriental Bank of Commerce and use a stock broker like Sharekhan.com?

A:

To start trading at a minimum you need 3 things;
Bank Account: This again is not must, but most preferred to transact. Quite a few broker would insist on this.
Demat Account: This is must as all shares on NSE are held electronically. The custodians are CSDL or NSDL both Government entities. These don't offer services directly to customer, but via other financial institutions like Banks and Large Brokers.
Broker Account: This is required to buy or sell securities. If you are only buying in IPO, this is not required as one can directly participate in IPO and Broker is not involved. However if you want to buy and sell on NSE you would need a broker account.
Quite a few financial institutes offer all 3 services or 2 services [Demat/Broker].
The fee structure and online service etc are differentiators. You can take a look at options and decide the best one to use.

