Index Investing is a passive way of investing . Having a bunch of stocks in portfolio exactly mirroring a stock in an index is a passive way of investing. This can be done by purchasing stocks mirroring an index and tracking the weight and readjusting the portfolio whenever the constituents or weightage in the index changes. Better is to buy an Index fund. This ensures that you get returns close to index.
Example if you bought an index fund of sensex when sensex was at 10,000 , it would have appreciated 70% when sensex is around 17,000. ( Returns would be exactly similar to the index minus tracking error , of a fund or an investor).
If someone believes that actively choosing stocks from the wide range of stocks available, invests in them then it is active investing.
In India most equity funds are actively managed funds. The index funds generally involves a lesser cost. Actively managed funds have managed to beat indices (on an avg.) many a times, excepting 2005-07 period.
Investing in either of the type of funds should be made by an investor after understanding the nature of products clearly.It's also not a bad idea to diversify your investments between active and passive funds.