For a non-ETF mutual fund, you can only buy shares of the mutual fund from the mutual fund itself (at a price that the mutual fund will reveal only at the end of the day) and can only shares back to the mutual fund (again at a price that the mutual fund will reveal only at the end of the day). There is no open market in the sense that you cannot put in a bid to buy, say, 100 shares of VFINX at $217 per share through a brokerage, and if there is a seller willing to sell 100 shares of VFINX to you at $217, then the sale is consummated and you are now the proud owner of 100 shares of VFINX. The only buyer or seller of VFINX is the mutual find itself, and you tell it that you "want to buy 100 shares of VFINX and please take the money out of my checking account". If this order is entered before the markets close at 4 pm, the mutual fund determines its share price as of the end of the day, opens a new account for you and puts 100 shares of VFINX in it (or adds 100 shares of VFINX to your already existing pile of shares) and takes the purchase price out of your checking account via an ACH transfer. Similarly for redeeming/selling shares of VFINX that you own (and these are held in an account at the mutual fund itself, not by your brokerage): you tell the mutual fund to that you "wish to redeem 100 shares and please send the proceeds to my bank account" and the mutual fund does this at the end of the day, and the money appears in your bank account via ACH transfer two or three days later. Generally, these transactions do not need to be for round lots of multiples of 100 shares for efficiency; most mutual fund will gladly sell you fractional shares down to a thousandth of a share.
In contrast, shares of an exchange-traded fund (ETF) are just like stock shares in that they can be bought and sold on the open market and your broker will charge you fees for buying and selling them. Selling fractional shares on the open market is generally not possible, and trading in round lots is less expensive. Also, trades occur at all times of the stock exchange day, not just at the end of the day as with non-ETF funds, and the price can fluctuate during the day too. Many non-ETF mutual funds have an ETF equivalent: VOO is the symbol for Vanguard's S&P 500 Index ETF while VFINX is the non-ETF version of the same index fund. Read more about the differences between ETFs and mutual funds, for