Phishing is any form of information gathering; generally associated with criminal activity. It is the process of getting private information. Phishing started making its heyday in the 1970's when the the first hackers started appearing. In order the break into secure networks (back then it was mostly just phone phreaking), hackers would have to go phishing for information.
The easiest way to go phishing is to dumpster dive. The perpetrator goes through the garbage looking for useful information (secret codes, manuals, etc...). Another type of phishing can occur on the phone. A conversation might go like this :
Random computer administrator: Hello, my name is Toby, how may I help you?
Phisher: Hey Toby, this is Jonathon Doe over in accounting, I forgot my password, can you reset it for me?
Kevin Mitnick (famous hacker) was a real pro at this - he called it social engineering. The modern variation is of course a fraudulant email asking for information; or in many cases a completely spoofed web page that tricks you into giving up your valuable private information.