Our flight was delayed from takeoff because the captain reported it to be "out of trim". What does this mean?
In this context, it would mean the aircraft is loaded in such a way that the Center of Gravity is too far forward or aft. That's actually not the way we usually use the word "trim" in aviation but it's what it means in this case.
"Trim" is the ability to correct for deviations in flight controls by prepositioning the flight surfaces for a particular direction. For example, if the aircraft is pitching up a bit by default, you can trim down in order to counteract it.
"Out of trim" would mean that the Aircraft has either been improperly trimmed, or is operating outside of the range trim can be applied. (If the aircraft is pulling down hard, you can only trim up so much.) This can be due to a mechanical defect, improper weight and balance, or other issues. He may not be able to safely guarantee the aircraft is trimmed in such conditions.
There is a related question here that explains it in further detail: What is an out-of-trim condition and how is it detected?
If the center of gravity was out of limits it is unlikely the crew would use the term "trim" to describe it. Also, if passengers needed to be moved to correct the condition you would see this happening.
More likely the fuel control unit was out of trim and needed to be adjusted. It is a fairly common procedure if an engine parameter is a little bit off from normal.
The field repair of the turbine engine fuel control is very limited. The only repairs permitted in the field are the replacement of the control and adjustments afterwards. These adjustments are limited to the idle rpm and the maximum speed adjustment, commonly called trimming the engine. Both adjustments are made in the normal range of operation. During engine trimming, the fuel control is checked for idle rpm, maximum rpm, acceleration, and deceleration. The procedures used to check the fuel control vary depending on the aircraft and engine installation.
The engine is trimmed in accordance with the procedures in the maintenance or overhaul manual for a particular engine. In general, the procedure consists of obtaining the ambient air temperature and the field barometric pressure (not sea level) immediately preceding the trimming of the engine. Care must be taken to obtain a true temperature reading comparable to that of the air that enters the engine. Using these readings, the desired turbine discharge pressure or EPR (engine pressure ratio) reading is computed from charts published in the maintenance manual.