The "basis" of an asset is what a person has paid for an asset together with money invested in an asset after purchase This determines gain/loss for income tax purposes.
A "stepped-up basis" occurs when assets get a new basis when they are passed by inheritance (through will or trust). These assets will be re-valued as of the date of death of the owner.
If the value of the asset has gone up in value since purchase, the new owner will take that asset with a new basis equal to the current value without paying tax on that increase. This has the potential to save a significant amount of capital gains upon sale in the future and can be a good way to do some strategic estate planning.
Example: My mom bought real estate in 1950 for $25,000. When she died last year it was appraised at $500,000. I received that real estate from her estate and my basis is now $500,000 tax-free. If my mom had sold this real estate before she died she would have had to pay taxes on $425,000 of gain, but because she died with the property I received a "step-up" in basis to the current value.