If you have any device on iOS 6.x, you can jailbreak it untethered using evasi0n. See jailbreaking iOS 6 guide for details.
If you have any device on iOS 5.1.1 (iPad 3rd generation, iPad 2, iPad 1, iPhone 4S, iPhone 4, iPhone 3GS, or iPod touch 3rd and 4th generation), you can jailbreak it untethered using Absinthe 2.0 or redsn0w. For more detailed advice, see jailbreaking iOS 5.1.1 guide.
If you have an older device on iOS 4.2.1 (iPhone 3G or iPod touch 2nd generation), you can also jailbreak it untethered using redsn0w. Plug your device into your computer, download redsn0w, open it up, and follow its instructions.
How do I jailbreak my old-bootrom iPhone 3GS untethered? Use the latest version of redsn0w to jailbreak it. It'll notice that your device is an old-bootrom device and will jailbreak it untethered on any version of iOS. To check if your bootrom is new or old, you can use @iH8sn0w's iDetector for Windows or f0recast for Mac.
Why am I having trouble jailbreaking my 8 GB iPod touch 3rd generation or iPhone 3GS model A1241 (A1324 China)? There's no such animal as an 8 GB iPod touch 3rd generation - you actually have an iPod touch 2nd generation - see this document. If you think you have an iPhone 3GS model A1241 (A1324 China), you actually have an iPhone 3G - see this document.
For other devices and other versions, check out the following chart to find the right tool for your device and iOS version. You can also get step-by-step jailbreaking guides with screenshots from iClarified, ModMyi, and iDownloadBlog(scroll down that page for the guides).
For a quick-reference guide to current jailbreaking tools, you can also check here: http://jailbrea.kr/
Tethered on 4.3, 4.3.4-4.3.5, 4.2.9-4.2.10, 5.0, 5.1, and 6.0-6.1 (untethered for old-bootrom 3GS users). Tethered jailbreaks not available for iPhone 4S, iPhone 5, iPad 2, iPad 3rd generation, or iPod touch 5th generation.