The iPod Nano, like all other iPods, has a 3.5-mm headphone jack. It comes with a pair of earbuds, and you can substitute any other headphones with the correctly sized jack. The most common problem with the iPod Nano headphone jack is that the headphones aren't working correctly. There are a few possible reasons for this, and it's easy to do your own troubleshooting before contacting an Apple Store.
Dirt and debris can clog the headphone jack and keep it from connecting properly to the headphones. Shine a bright light on the headphone jack and look closely for any debris. Blow gently to remove any dirt, and pick any larger pieces out with a toothpick.
Look at the cord of your headphones for any kinks, holes or exposed or broken wires. The earbuds that come with the iPod Nano usually don't last much more than a year of regular use, especially if they're regularly bumped and dropped. Try a different set of headphones to see if that fixes the problem.
Sometimes the problem with the jack is actually in the software, not the hardware. Reset the iPod Nano by holding down the Sleep and Volume Down buttons until the screen goes dark and the Apple logo shows up. This usually fixes any minor software problems.
When none of the above steps work, it's possible that the headphone jack is damaged or malfunctioning. Your iPod Nano came with a one-year limited warranty. Contact Apple to ask about repair or replacement.