"Stepped-up basis" refers to a tax rule that minimizes or eliminates capital gains tax liability. Say, for example, your Uncle Buck owns Apple stock. He purchased 100 shares of the stock when it was worth $1/share. In tax lingo, his cost to buy the stock is known as his "basis." Apple stock is now worth about $170/share. If Uncle Buck sold his shares today, he would have to pay a capital gains tax on the $169/share appreciation in the value of his stock. Similarly, if Uncle Buck gifted you the stocks today, you would "inherit" his basis, meaning you would have to pay capital gains tax on the $169/share appreciation if you sold the stock tomorrow.
But let's say Uncle Buck decided to hold onto his shares. He placed them in a trust that names you as the sole beneficiary. If Uncle Buck died today, leaving you the stocks, the Internal Revenue Code provides that the value of the stock today (Uncle Buck's date of death) is your new basis in the stock. In other words, your basis is "stepped-up" from $1/share to $170/share. So if you turned around and sold the Apple stock for $170/share, you would pay no capital gains tax.