The TIE Fighter is the primary fighter in the Imperial Fleet. It's a shortrange fighter not capable of traveling through hyperspace.
As everyone knows, Darth Vader flew a TIE Fighter with a slightly advanced design during the Battle of Yavin. Additional wing panels were made for this model to simulate the battle damage that Vader received when Han cleared the Death Star trench.
Introduced in Return of the Jedi, TIE Interceptors are supposed to be a more advanced version of the TIE Fighter.
The Imperial Shuttle was based on a design for the skyhopper, a vehicle that never in flight on screen. The designed was borrowed for this elegant vehicle used by the Empire.
For a little bit of Fettspoitation...check out Slave I! The design for this ship was based on a street lamp. This model was only used for the filming of The Empire Strikes Back.