What protoss unit can transport others?

To physically transport Protoss you need the Warp Prism built in a Robotics facility. The Robotics facility require a Cybernetics core to be built, which require a Gateway (which require a Nexus) to be built.

However Protoss have a better way to "transport" new unit to the battle field : A Gateway can be transformed into a Warp Gate (The research to be able to do that is in the Cybernetics core). A Warp Gate is able to create a new unit anywhere in the psionic matrix.

Psionic matrix is the blue area around pylons. A common tactic is to create a "proxy" pylon near the front line of the battlefield, in such a way that the pylon is near enought to be able to transport unit quickly near the enemy but far enough to not be spotted by enemy forces.

Moreover the Warp Prism (mentioned earlier) is able to go into phase mode so he can act as a pylon (and have is own pysionic matrix) but is unable to move while he is in this mode.

In addition to what others have mentioned (Warp Prism, warping directly), the Mothership is also capable of transporting other units, in its own way - via Mass Recall.