I suspect you have have a 2G iPod. Those can only go to iOS 4.2.1.
Identifying iPod models
iPod touch (3rd generation) features a 3.5-inch (diagonal) widescreen multi-touch display and 32 GB or 64 GB flash drive. You can browse the web with Safari and watch YouTube videos with Wi-Fi. You can also search, preview, and buy songs from the iTunes Wi-Fi Music Store on iPod touch.
The iPod touch (3rd generation) can be distinguished from iPod touch (2nd generation) by looking at the back of the device. In the text below the engraving, look for the model number. iPod touch (2nd generation) is model A1288, and iPod touch (3rd generation) is model A1318.
To update
The Settings>General>Software Update comes with iOS 5 and later.
Connect the iPod to your computer and update via iTunes as far as your iPod model allows
iOS: How to update your iPhone, iPad, or iPod touch
A 1G iPod can go to iOS 2.2 via iTunes and iOS 3.1.3 via
Purchasing iOS 3.1 Software Update for iPod touch (1st generation)
https://buy.itunes.apple.com/WebObjects/MZFinance.woa/wa/touchLegacyLandingPage
A 2G to 4.2.1. Requires iTunes version 10.X. If a Mac it requires OSX 10.5.8 or later.
A 3G to 5.1.1 Requires iTunes version 10.5 or later
To more easily find compatible apps:
iOSSearch - search the iTunes store for compatible apps.
Vintapps 3.1.3 - paid app.
Apple Club - filter apps by iOS version.
Discussion post by msgarmar - technique to sort apps by date.