Trim:
Control surface trim (aileron, rudder, elevator/stab trim) is a function of every aircraft I know of (small GA to large airliners) to help relieve control forces on the control column during a manoeuvre that requires the stick be moved away from the central point.
For example, you are taking off from a runway, you reach rotate speed and you rotate, and hold the attitude to climb away. As you're at a slow speed, controls aren't particularly responsive and more often than not, you'll have to apply a bit of force to hold the yoke in a rearward position to keep the climb going. After a while, it starts to take its toll - you start getting distracted by other things, you are looking outside, talking to ATC - you gradually start reducing or increasing the quite specific force that you're keeping, away from the ideal force. The aircraft reacts, the nose will drop, climb rate reduces, speed increases, the pilot must then recover this (or vice versa). Thus, you have flying inaccuracies. On a simpler level, it's just a pain to have to hold a control force for a long time, especially in the cruise.
To that end, trim (elevator trim in this case) is used to relieve control forces by moving a wheel to allow the control yoke to go back to its natural centre whilst maintaining the climb. In theory, when you fly in trim, you can relieve the force to the point that you can let go of the control column and the aircraft will maintain the attitude as though you still had the force applied.
Rudder and aileron trim are slightly different in that they're not so widely used - in fact a lot of small GA aircraft don't have rudder / aileron trim. Rudder trim is used to keep the aircraft in balance and alleviate the force applied by the foot on the rudder. The biggest example of this is during an engine failure on a multi engine aircraft. You control the aircraft and asymmetric thrust (resulting in yaw) by using the rudder alone. In propellor aircraft especially, it's a hell of a force you have to apply and in this case, trim is a must because your leg soon goes dead! Trimming the force out moves a tab on the rudder holding in the force you were applying.
Aileron trim is even less likely to be used but could be used in the same scenario of an engine failure, since adding 5 degrees or so of bank into the remaining live engine can increase performance marginally. Or perhaps you have a fuel imbalance you want to correct with aileron trim.
As for Trim Air...never heard of it. Are you sure you've got the term right?
If you want an explanation of the other items you say you've found, just ask.