When you inherit property, such as a house or stocks, the property is usually worth more than it was when the original owner purchased it. If you were to sell the property, there could in theory be huge capital gains taxes. Fortunately, when you inherit property, the property's tax basis is "stepped up," which means the basis would be the current value of the property.
For example, suppose you inherit a house that was purchased years ago for $150,000 and it is now worth $350,000. You will receive a step up from the original cost basis from $150,000 to $350,000. If you sell the property right away, you will not owe any capital gains taxes. If you hold on to the property and sell it for $400,000 in a few years, you will owe capital gains on $50,000 (the difference between the sale value and the stepped-up basis).
On the other hand, if you were given the same property, as opposed to receiving it upon the owner's death, the tax basis would be $150,000. If you sold the house, you would have to pay capital gains taxes on the difference between $150,000 and the selling price. The only way to avoid the taxes is for you to live in the house for at least two years before selling it. In that case, you can exclude up to $250,000 ($500,000 for a couple) of your capital gains from taxes.