Index funds are mutual funds that attempt to mimic the performance of a particular benchmark, or index, by buying and holding the same stocks found in that index. Index funds are passive investments, meaning that stocks in an index fund are not bought and sold regularly (“actively managed”); essentially, index funds do not try to beat the market – they try to mirror their benchmarks. Index funds work on the principle of, if you can’t beat ‘em, join ‘em”.
There are index funds available that track all the various indices, including the major ones such as the S&P 500, the Nasdaq Composite, the Wilshire Total Market Index, or the Russell 2000; there are also sector-specific, country-specific, and all sorts of other niche index funds that track very narrow benchmarks. 
Traditional index funds have a longer history than ETFs, the former having been around since 1975 and the latter only since 1993. Index funds are still more widely held investments than ETFs, though the popularity of ETFs is growing: by 2004, the total net assets in ETFs were half those found in traditional index funds  and we imagine that, today, this ratio has increased further. Expense ratios for index funds are, for the most part, between 0.2-0.5%, and (usually) index funds usually have no brokerage fees (called “loads”) since you generally buy index funds directly from the mutual fund company or your brokerage house. Charles Schwab, in particular, built their brokerage franchise in no small part by offering no-load mutual funds.
An exchange traded fund, or ETF, trades like an individual stock. Just like a traditional index fund, it is passively managed and represents a basket of stocks that follows an index like the one’s listed above. However, unlike a mutual fund or an index fund whose Net Asset Value (NAV) is quoted at the end of the day, and ETF prices are quoted in real time just like a stock and you have total control over when and at what price you buy or sell. Index ETFs can be bought, sold, held, short sold, or bought on margin just like individual stocks, and brokerage fees must be paid per transaction as well as a low annual asset based fee.
The two main advantages of an index ETF over a traditional index fund are that ETFs are generally cheaper and definitely more flexible. ETFs are loved for their trading flexibility, as you can buy and sell them at any time during the trading day rather than at market closes, their exemption from the short sale uptick rule (a rule that prevents investors from shorting a stock unless the last trade of the stock resulted in an increase in stock price), and the fact that expenses for index ETFs are typically lower than those associated with traditional index funds.
Index ETFs do have one downside: because they trade like stocks, they are subject to the same buy and sell trading commissions as regular stocks. Many brokerage houses will let you invest in and sell positions in traditional mutual funds without paying any commissions whatsoever. If you are actively trading your Index ETFs, the brokerage fees can be meaningful and can take a significant bite out of your returns. However, Index ETFs have lower minimum investment amounts than traditional index funds, as you can buy a single share as opposed to investing $1,000 or $2,500 to get started.