Which kinds of tooth movements can be done in orthodontics?

Many. Teeth can be moved in all three planes of space through orthodontics. They can be moved up and down, rotated and side to side. Orthodontics will allow your teeth to be aligned properly so that they will look great and function properly.
Many. The better question is what can't be done. Teeth can be shifted and turned and moved in many directions and angles. Some cases, however, such as extreme underbites in adults, may require surgery and sometime we need teeth to be pulled if crowding is extreme. But generally, most of what you can imagine can be done with orthodontics. Some have even called our work 'miracles'!
Three major. Teeth move in three dimensions: tipping, rotation and translation.
Oh so many things. Given adequate bone support and good jaw relationships, there's so much that can be done. And if jaws don't relate well, that too can be corrected. In the hands of a qualified Orthodontic Specialist expect 90-95% improvement for children, 85-90% improvement for adults, with traditional Orthodontic treatment. See an Orthodontic Specialist to review your unique problem and treatment solutions.
Many tooth movements. Orthodontics can move teeth in many different directions. Teeth can be rotated, bodily moved, intruded, extruded, tipped and torqued. It all depends on what is needed.