When Shepard boards the Flotilla, she's greeted as "Captain Shepard," which she corrects by saying she was never made a captain, but only a "commander." What exactly is the "commander" military rank? Is it just kind of a catch-all term for someone in charge of a specific group or vehicle?

Commander is the naval rank under Captain.

captain and commander far side of the world was an ok movie

^Correct.
 
Shepard is still referred to as Commander Shepard, because she wasn't made a Captain by the time she was inducted into the Spectres.  The Normandy ceased to function as an Alliance vessel in practice, and became her personal flagship.  That the quarians refer to her as Captain has more to do with their cultural mores.

Commander is the rank just under Captain. Usually, the Commanding Officer of a naval ship would be a Captain, while the Executive Officer would be a Commander. As I understand it, the XO takes care of the day to day running of the ship so that the Captain can focus on more important matters. If you know your Star Trek: TNG, Commander is the rank Riker holds.

Also worth knowing is that when there's no superior officer assigned aboard, or alive, the highest ranking officer becomes acting captain while still not having the actual rank. 

SHEPARD!

"Captain" is both a rank and a post, at least in the US and Royal navies - it's logical to assume the same rules apply in the Mass Effect universe, but they don't necessarily have to.  Whoever is in permanent command of a ship, regardless of their rank, is the captain of that ship.  Typically, people commanding ships also hold the rank of captain, since such an important post is usually associated with senior officers.  On smaller ships, however, junior officers are sometime captains.  For example, in WWII, Kennedy was the captain of PT-109, a dinky little patrol boat, but his rank was only Lieutenant (Ensign -> Lieutenant JG -> Lieutenant -> Lieutenant Commander -> Commander -> Captain -> Admiral). 
 
So Shepard's rank is Commander, but Shepard's job is captain of the Normandy.  The Normandy is fairly small, just a frigate, so it makes sense that she would be commanded by someone who wasn't a full Captain.  Of course, the whole thing is weird, since Shepard is also a Spectre, working for Cerberus, and probably not technically on active duty with the Alliance Navy anymore.  Whatever, dude. SPACE.

I do hope he makes Admiral in ME3. 

Is an Admiral the equivalent of a General?

" I do hope he makes Admiral in ME3.  "

" captain and commander far side of the world was an ok movie "