A share of stock is a piece of paper covered with fancy embossing and pictures of eagles and things. The way you sell a share of stock is, you and I meet under a buttonwood tree, and I say I will pay you 10 gold dollars for one share of Amalgamated Pennyfarthings stock, and we shake hands, and then a day or two later you have your clerk bring over the piece of paper to my clerk, and I have my clerk bring your clerk the 10 gold dollars in a little bag labeled "$," and your clerk bites my gold dollars to make sure that they contain the right amount of real gold, and the clerks make the exchange, and I put the fancy piece of paper into my vaults and bask in my new part-ownership of Amalgamated Pennyfarthings.
Or that is sort of how it used to go, and it is also sort of how it notionally goes today. But the way it actually goes today is that you go to your online brokerage account and put in a market order to sell one share of Universal Hoverboards, and I put in an order to buy one share, and a computer matches us up, and then my broker automatically adds one share of Universal Hoverboards to the electronic file representing my account, and your broker automatically subtracts one share from your electronic account, and vice versa with the money. And then a thing called the Depository Trust Company, which is owned jointly by your broker and my broker and all the other brokers, does the same thing for the brokers, subtracting one share of Universal Hoverboards from your broker's electronic account and adding one to my broker's electronic account, at the brokers' electronic instructions. There may or may not be an embossed piece of paper -- probably not -- but if there is, it sits in DTC's vaults and no one's clerk needs to move it anywhere. All of this could happen, in round numbers, instantly, though in practice it mostly takes three days.