I'll try to explain a little better.
Standard Gear: Transmission has fixed gears and the tractor must come to a complete stop to shift to any gear.
Synchronized Transmissions: Allows for shifting on the move with the use of a clutch pedal. The transmission normally has some type of sychronizer that aligns the slines between the gear you are leaving and the gear you are going too.
Manual Synchro Shuttle Trans: Is similar to the above transmission with a lever to chnage direction forward and reverse. Minimum of two levers one to select the speed(gear) and one to shuttle. YOu push the clutch to shift forward and reverse and push the clutch to change speeds.
PowerShuttle: Instead of the clutch this a a hydraulic system that allows forward and reverse shuttling without the need to clutch. allows for rapid direction change. Speeds are normally still Synchronized and require clutching to change speeds or on some models and version the shuttle lever in neutral or like the MF 3 Function Shuttle on large tractors has a declutch feature by just lifting the lever and speed changes can be made.
Hydrostat: is infinite speeds by pushing pedals Forward or Reverse harder you push faster it goes. Hydrostats are great in smaller tractors but they require alot of horsepower to operate.
You might also check out
Massey Ferguson : Tractor Advisor : Home look in the section Tractors 101 they have some great basic knowledge of tractors too.