JailbreakMe

This was the exploit used to easily jailbreak 1.1.1 phones.

Credit
metasploit, rezn, dinopio, drudge, kroo, pumpkin, davidc, dunham, planetbeing and NerveGas

Exploit
There was a buffer overflow in the iPhone's libtiff. This was exploited to run a small application to jailbreak and patch libtiff. This exploit was also used in the PSP.