For those of you filing California state tax returns, the short answer is "probably." If you bought something and didn't pay sales tax on it, then you owe use tax on your tax return.
For the longer answer, consult your tax return form instructions or the State of California's Franchise Tax Board website. Or if you lack the patience for that, here's my rough understanding of it, as a regular California resident filing as an individual (so none of this applies for businesses or crazy high-rollers):
Use tax is all that "missing tax" that you didn't pay during the year on purchases from online retailers that didn't collect sales tax from you. Use tax has actually been obligatory in California for decades, originally imposed on purchases made out of state, but the state didn't emphasize it until Amazon.com started making billions while collecting no sales tax. Since 2003, California tax forms (540/540A) have included a line for you to report your use tax. Of course, quite a lot of people still didn't report what they owed in use tax, because it was basically an "honor system;" the state had no realistic way to track your online shopping, so it couldn't enforce anything (despite its ongoing threats even to go back and collect interest on unreported use tax from the last eight years). Moreover, it actually asked consumers themselves to manually track all their own purchases and calculate the use tax, which was a huge pain.
In 2012, California finally tried to make it a lot easier by devising the "Estimated Use Tax Table," which allowed consumers to simply pay a specified amount determined by their income range. For example, if you made less than $20,000 in 2011, then your estimated use tax would be a mere $7, even if you had actually dodged more than that in sales tax by shopping online. If you legitimately owed less (because you didn't do a lot of online shopping, in which case good for you), you could, of course, still report less. You also still had to manually track and calculate for anything that cost over $1000, but, again, it was the honor system.
This year (2013, filing your 2012 tax return), use tax is reported on line 95 of Form 540/540A, and the Estimated Use Tax Table is back again:
In late 2012, however, the state finally took away Amazon's tax-free advantage in California, which had been the biggest cause of the "missing taxes." So it's actually possible that, moving forward, you'll owe a lot less use tax, or even none at all (unless you can find another online retailer that won't collect sales tax in California).
(In case it's not clear, the above only applies to California, because that's where I reside and file. There are other states that also collect use tax, but I wouldn't know anything about how they handle it.)