Tethered means you have to rejailbreak with every reboot, including when you lose power. This has been an issue since October 2009 when the new bootrom iDevices (ipod touch and 3GS) were released. Apple changed the hardware midway through production. Old bootrom 3GS' were not affected.
The iphone 4 4.2.1 jailbreak is only a partially tethered jailbreak, though. Your phone will still boot; it will just boot stock (until you run the jailbreak tool).
The iPhone 4 4.1 (and earlier) firmware jailbreaks are untethered. If you buy an iPhone that's running 4.2.1, you will not have an SHSH and you will not be able to unlock or have an untethered jailbreak, for now at least.
Nice! What a great set-up having one as a Sunday driver and another for the track only. Tell your dad I'm *very* jealous. Would he ever let you, or a (cough, cough) friend borrow one? Just for a little while?