Catapult
An egg launcher designed in the style of a catapult depends on a launching arm propelled by a spring or other source of tension that can be released suddenly. An advantage of catapult-style egg launchers is their potential for high accuracy and consistent range once properly calibrated. Build a frame, launching arm and either a spring that pushes the launching arm up or a crank that uses the arm's own tensile strength to build enough pressure for a launch. The frame should stop the arm at a certain angle after the trigger releases it.
Trebuchet
A trebuchet is similar to a catapult, but instead of pressure on the launching arm, it is driven by a counterweight that drops suddenly and lifts the launching arm swiftly into the air. Usually a trebuchet has a flexible sling at the end of the launching arm that snaps up to add more leverage to the launch. Trebuchet-style egg launchers have a relatively long range but may be harder to launch with consistent accuracy.
Cannon
An egg launcher in the style of a cannon or bazooka launches the egg through a tube by building up pressure behind the egg and releasing it suddenly. The pressure may come from a chemical reaction or an air compressor. Cannon-style egg launchers have the advantage of a very long range and the potential for great accuracy when they are designed and calibrated properly. Be careful, though, as the high-pressure launches or explosive reactions involved make this style more dangerous to construct and operate than other models.
Slingshot
Slingshot-style egg launchers use the tensile strength of a flexible band to launch the payload. The band may be elastic or just a slightly stretchy material under high tension. Slingshot launchers that aim for consistency should have a frame that holds the band at both ends and in the middle. The part that holds the middle should be a trigger that can pull the band back to a point that can be calibrated for range then released to launch the egg. Slingshot launchers are relatively simple to build but may have a lower range and accuracy than other styles.