The root part is the first part placed in the VAB or SPH so it's on the top of the dependency tree. A part requires at least one attachment node and may not be a physicless part to be able to be used as a root part. Prior to version 0.20 the root part also needed to be a command module. Since then, it is possible to have craft without any control authority, but the game warns about that problem upon an attempt to go to the launchpad or runway.
If a part is selected, it will select all parts placed onto it; therefore if the root part is selected, it will always select the entire craft. Because it's not attached to another part, selecting it won't allow the player to reattach it to another part.
It is possible to change the root part of an already built construction in the VAB/SPH using the Root icon (fourth from left) of the assembly mode selector icons (Editor Gizmos). In this mode the part clicked on becomes the root part if it satisfies the conditions of being the root part.
The root part is always placed by only clicking on the part, so it's not possible to place it onto anything or with a symmetry. Since 0.90 it is possible to replace the root part, using the new Editor Gizmos. Before that all parts directly attached to the ship needed to be disconnected in order to delete the root part and select a new one. It wasn't possible to use an already placed part.
Because symmetry is relative to the root part, symmetry behaves differently if other parts are placed on already symmetrically placed parts. In this case one part is placed on each symmetric parent part.
When a craft separates into multiple vessels, the craft containing the original root part will become the active vessel. The part highest in the dependency tree in each vessel becomes that vessel's root part. It should be noted that the camera actually focuses on the center of mass of a vessel while in flight, not its root part.
Although it's not possible to build a craft out of only physicless parts (because a craft requires a root part and those can't be physicless), it is possible to get a craft later (via in-flight separation) which does consists only of physicless parts. This was fixed in 0.18.2 so that the new root part would then be forced to obey physics, regardless of its "physics_significance" value.