"Just got selected to be a Naval Aviator & Naval Flight Officer!
When you submit you're [sic] application, you put down which selection boards you want it to go to. You can select just pilot if you wish. My order of preference was 1. Pilot 2. Naval Flight Officer (Goose) 3. Surface Warfare Officer. All 3 of those designators meet for the same selection board. So, if I don't pass medical for Pilot for something like vision, since I was also accepted to my #2 choice, I will be designated as NFO instead of completely out of the Navy."

The character nicknamed "Goose" in the film Top Gun is an NFO, acting specifically as a Radio Intercept Officer (RIO).
https://en.wikipedia.org/wiki/Top_Gun

I don't know whether "goose" was already a colloquial navy term for an NFO when the film was made or whether it's derived from the film (quite probably the latter), but that's what it is.

Definitely not a trainee of any kind, and not a pilot, which is called an aviator in the navy. It's specifically US Navy and Marines. The USAF equivalent is a Combat Systems Officer (CSO); I don't know a CSO is also known as a goose.

An NFO is not a pilot, though may perform co-pilot functions. They basically handle weapons systems, electronics and navigation.

"Naval Flight Officers operate some of the advanced systems on board most multi-crew naval aircraft, and some may also act as the overall tactical mission commanders of single or multiple aircraft assets during a given mission. NFOs are not formally trained to pilot the aircraft, although they do train in some dual-control aircraft and are given the opportunity to practice basic airmanship techniques. Some current and recently retired naval aircraft with side-by-side seating are also authorized to operate under dual-piloted weather minimums with one pilot and one NFO. However, in the unlikely event that the pilot of a single piloted naval aircraft becomes incapacitated, the crew would likely eject or bail out, if possible, as NFOs are not normally qualified to land the aircraft, especially in the carrier-based shipboard environment.
NFOs serve as weapon systems officers (WSOs), electronic warfare officers (EWO), electronic countermeasures officers (ECMO), tactical coordinators (TACCO), bombardiers, and navigators."
https://en.wikipedia.org/wiki/Radar_Intercept_Officer#Naval_...

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Note added at 9 hrs (2014-08-27 11:31:38 GMT)
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I forgot to provide the source of my first reference:
http://www.ign.com/boards/threads/just-got-selected-to-be-a-...

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Note added at 1 day10 hrs (2014-08-28 11:36:11 GMT)
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See discussion for further details on this, including evidence that US Navy and Marines training involves up to about 200 hours flight time, so someone with 600 hours is certainly no longer a trainee.

Here's a reference which, if reliable (and I see no reason to doubt it), resolves the meaning and origin of the term. It's about the author Zachary J. Kitchen:

"He always dreamed of flying so, after graduating with a B.S. in English Literature from the Academy, he went to flight school and flew for the Navy for five years as an NFO or "goose" (as the position has been known since the film "Top Gun.")"
http://www.amazon.com/Zachary-J.-Kitchen/e/B00CG9LHZU



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Note added at 1 day10 hrs (2014-08-28 11:43:54 GMT)
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It makes sense. "Top Gun" was a very influential film, in that it made a lot of young men want to enlist and fly US Navy planes. There are many web pages that explain what an NFO is by saying "think of Goose in the movie Top Gun".