Your grandparents might still file their taxes with paper forms. You probably e-file.
First, grandma might spend 20 minutes looking for her glasses, only to discover they were dangling around her neck on a chain all along. Second, she gets out the tax forms she picked up at the library or the post office. Then she gets out a big shoe box full of receipts. Finally, she sits down with the forms and the receipts and a pencil and a calculator (or maybe an abacus, if she's going really old school).
After a while, she finishes filling out the forms and stuffs them into an envelope. Finally, she drives to the post office to mail it, where she'll spend 20 extra minutes holding up the line while she complains about the price of stamps nowadays.
That process represents paper filing. The alternative is e-filing.
The e-filing method involves letting a computer do most of the math for you. Then, when your taxes are done, you click a button and it gets sent electronically to the IRS and the appropriate state tax authorities. Then, if you want to complain about stamp prices, you can just do it on Twitter.