Today, a demat account is a must if you want to buy or sell equities. Of course you can trade in futures and options without a demat account as they are cash settled products anyways. However, if you want to invest in equities either through IPOs or through the secondary market then you need a demat account. In fact, you can hold not just equities but also other assets like bonds, gold bonds and even mutual funds in your demat account.
As an investor you can open a demat account with any registered depository participant (DP). More often than not, your broker will also be a DP; and it is always better to have the trading account and the demat account at the same place to facilitate seamless and hassle-free debits and credits to your demat account. But the big concern for a lot of small investors has been the costs associated with demat. You are right that you don’t need a demat account when your trading intraday but the SEBI regulations stipulate that you cannot trade equities without a demat account. There are two reasons for the same. You may have bought the shares for intraday but you may not be able to sell the shares because of illiquidity. As a result you need to take the delivery. Sometimes, you just forget to close your intraday trade and although the broker will cut all open intraday positions by around 3.15 pm, in case the broker system fails on that day, the onus is still on you. That is why SEBI insists on having a demat account for trading equities, even it is only for intraday.