Okay. There are really two choices here, not 3.

• Mutual Funds
• Exchange-Traded funds

The two are the same, except for how they're bought/sold.

• Mutual Funds are bought by the dollar, based on a price calculated based on the value of all the components shortly after close of market.
• Exchange Traded funds are bought by the share, and are traded like stock. In theory you can snag a lower price by using stock trading techniques. But you can only buy in units of shares... So If the ETF is trading at 151.66 and you have 200,000 to invest, then you can buy 1318 shares of the ETF, leaving behind $111.88.

Index funds are just a sub-speciality of Mutual Funds (or, frankly, of ETFs)

Funds can be:

• actively managed.
  They hire smart people to research and watch and choose what to invest in. The manager might have heard of supply issues on Apple's part, and might be seeing that Toro's weed-eaters are selling extraordinarily well...and adjust their buying plans to match. But with that expertise comes extra cost. Expense ratios tend to be higher, and there's "loads" which are nothing more than trading fees.
• indexed.
  These follow a pre-defined index, like the "Dow Industrials" or the S&P 500. But there's lots of different indexes out there. The advantage is that it's cheap and easy for the manager to manage. You put your 200,000 in, the index tells them exactly how much of Apple, how much of BP, and how much of Toro to buy.

Both can be indexed.