A mutual fund is a collection of different stocks chosen by some management company that get packaged together and sold as a unit. Traditionally, orders for mutual funds go through at the end of a trading day, and the price is based on the closing price of the underlying assets.
An index fund is a mutual fund that tracks a particular index (the S&P 500, for example) as opposed to selecting stocks based on the manager's choice.
An exchange traded fund (ETF) is a mutual fund that trades throughout the day on a stock exchange, as opposed to only at the end of the day.
Many of Vanguard's funds are available as both a traditional mutual fund and an ETF. The underlying assets are identical. Whether you prefer to buy the traditional fund or the ETF is up to you.