First try the headphones in another device but first look at the plug for abnormalities that could damage the headphone jack in the iPod.
If confrmed that the headphnes work OK
- Try cleaning out/blowing out the headphone jack. Try inserting/removing the plug a dozen times or so. There is a little switch in the headphone jack that disconnects the speaker when the headphone jack is inserted.
Try the following to rule out a software problem
- Check also Settings > General > Accessibility > Hearing. Make sure that the sound balance is not set all the way to "R".
- Reset the iOS device. Nothing will be lost
Reset iPod touch: Hold down the On/Off button and the Home button at the same time for at
least ten seconds, until the Apple logo appears.
- Reset all settings
Go to Settings > General > Reset and tap Reset All Settings.
All your preferences and settings are reset. Information (such as contacts and calendars) and media (such as songs and videos) aren’t affected.
- Restore from backup
- Restore to factory settings/new iOS device.
- Make an appointment at the Genius Bar of an Apple store. Seems you have a bad headphone jack.
Apple Retail Store - Genius Bar
If not under warranty Apple will exchange your iPod for a refurbished one for this price. They do not fix yours.
Apple - iPod Repair price
A third-party place like the following will replace the jack for less. Google for more.
iPhone Repair, Service & Parts: iPod Touch, iPad, MacBook Pro Screens
Replace the jack yourself
iPod Touch Repair – iFixit
