Skins that affect game mechanics

Skins have absolutely no effect on your damage output/resistance, attack/projectile/movement speed, or on-hit effects.

In a nutshell, all a skin does is change the appearance of your champion. There are some cases (Legendary skins, for example) where the skin will have a more significant effect on your champion. This can come in the form of skills looking different while keeping their original effects, champion voices changing, or the way your character walks.

Some (but not all) examples:

• Frostfire Annie - Blue flames instead of orange.
• Gentleman Cho'gath - A more civilized English accent, as well as some altered dialogue
• Woad Ashe - Different movement animation (still maintaining default movement speed)

Currently, the only champion whose skin affects other champions (in addition to his own appearance) is Pulsefire Ezreal. However, even this is a minor change, changing the enemy champions'/minions' death animations.

The case that you mentioned regarding Sunglasses wearing champions in relation to Leona is a rare case. Off the top of my head, it is the only case (I could be wrong, though), and even then the damage reduction is utterly minuscule.

For additional reference, the Skin Spotlights channel on YouTube has a fairly comprehensive list of videos showing various and many champion skins.