Generally, Stocks move the index. Can index also move the stock?

Per definition not - the Index is only a published number and when you add the price of certain things and report this number, the number can not change the price of the items.

Now, if you have financial products that trade of the price of the index - futures, ETF - then yes, someone going long a large amount WILL change the index because unless someone sells him for speculation, a market maker will enter the short (vs the long of the investor) and offset the risk by - buying the underlying instruments or a basket that closely behaves like that.

But technically, this is not the INDEX changing and driving the market - it is a product BASED ON the Index

The index is an average of the stocks in the index. A very simplistic index consisting of 3 stocks might be calculated as:

(share 1 price + share 2 price + share 3 price) / 3 = index price

In practice the calculation will be more complicated, but the principle stands that you put in a set of share prices and get out an index price. By that definition, the index price does not directly affect the price of any share.

However, the fact that a share is in an index, and the performance of that index may affect investors choices when they decide on their trades.

For example, the FTSE 100 is an index of the 100 most valuable UK companies. Every three months the list is reviewed, and some companies will be removed, while other will take their place. Any fund that is tracking the FTSE 100 will need to buy shares in the incoming companies, and sell shares in those that have dropped out. This causes a rise in the price of a share that is likely to enter the index, and a fall in the price of a share about to leave it.

Several answers mentioned that derivatives of the index can move the stocks but none provided examples so I though that I'd offer one:

The price of index futures equals the underlying index value only at expiration. Prior to that, they reflect the future price of a stock index given pending dividends and current interest rates. The long futures trader does not receive dividends from the index stocks whereas the owner of those stocks do.

When index futures prices deviate from the fair value of an index, it presents an opportunity for arbitrage via buying or selling of the stocks.

If the index futures price moves far enough away from fair value, institutional traders will step in. If the futures trade at a premium, they will sell the index futures and buy the component stocks. If the futures trade at a discount, they'll buy the index futures and sell the component stocks.

As others have mentioned, the index itself is an imaginary construct derived from the prices of its constituent stocks. Therefore the index can't be changed externally and therefore have an effect on those stock prices.

However, as also mentioned, there exist actual commercial products such as ETFs which actually buy stocks in an attempt to simulate indices. These products are priced independently of the index itself (but obviously tend to be highly correlated to it).

The existence of these products allow an index to indirectly impact its constituent stocks in myriad ways, but usually not significantly more than other stocks which are correlated to the index (but don't belong to it).

One exception, however, is moments where the index changes, either adding or removing a company from its ranks (or changing its weight in the index's calculation). When this happens, ETFs are forced to realign their portfolios to match the index in relatively short order. And since ETFs are a massive market, these changes to the index can apply dramatic pressure to the affected stocks as they are bought or sold by the ETFs.