I have played Leona for going on 7 seasons now, and have found her to be consistently the strongest champion in this game. She can be used as a frontline tank, a strong peeler, a back line diver, and everything in between. Her massive CC allows her to completely control at least one enemy in every fight.

The Basics:
An extremely important part of any champion is knowing the little things that makes them strong, the best known of these being Riven's auto attack cancelations. The most important hidden strength of Leona is her unique auto attack reset on her Q. If she uses Q to empower an attack, and the player clicks again for another auto attack (you MUST click), the extra auto does not reset its timer like that such as Nasus. This allows a good player to nearly insta kill wards placed near them. Simply click to auto, use Q for a second hit, and click again for the 3rd hit, which will land as though the q had no been cast. This combo should also be used in most cases when engaging onto champions with E, though more details on that later.

Pros:
Extremely durable with W.
Very mobile in terms of engage.
Long range engage with R, once the player is used to the cast delay.
AoE cc.
Can 1v1 MOST champions past level 11.

Cons:
Not very mobile in terms of disengage, but with stuns can often escape.
Can often abandon her carry, as once she enters into a fight it is a slow process to leave it.

Always max W, period. The extra defenses and large burst of damaged, coupled with Aftershock, is required.

Secondly, many players will disagree with maxing Q second, as it provides no cooldown benefit. However, in the case of playing as an engager, the extra damage with sheen is needed to fight down carries. This said, if the enemy team is tank- or engage- heavy (as in you will be acting almost entirely as peel) it is better to max your E first, as you will not be able to hurt the tanks that you are trying to keep off your adc.

In 80% of games, 2 levels of E and then maxing Q is the best choice.

Aftershock is must, no exceptions.
Unflinching is my preference as the longer you are cc'd the less cc you can use, since your Q's cooldown is so short at max CDR.
Conditioning is the best choice since Leona will not need extra defenses early; her W will cover that.
Overgrowth is another preference; second wind can be used if you suck at dodging spells or staying just outside of auto range.

I strongly recommend sticking to my secondary runes unless you plan to purchase more than 30% CDR (your Eye of Equinox and Iceborn cover this)
The extra ultimate cdr places you R on just over a 30 second CD at level 16, allowing it to be used EVERY fight, and also every minor fight in between.

The dream would be to first back a sheen, every game. Leona's high base AD makes an early sheen win most lanes. You will likely be pinged for buying this early, so be sure to get pink wards to make up for your lack of sightstone, which should be your next purchase along with tier 1 boots. In 75% of games merc treads is the choice, as few magic resist (MR) items are good on Leona. If the enemy team is AD heavy with little cc, the armor boots is the better choice. Mobi boots are mostly useless as you are ALWAYS dealing or taking damage, and we easily max cdr without Ionian boots. Finish the sightstone into Eye of Equinox, followed by sheen into iceborn. Don't be tempted to finish iceborn first, the health from Equinox makes Leona extremely deadly in the start of the mid game. Once Equinox+Iceborn+Merc treads are done, your core build is complete. With only these 3 items Leona can win most games, after this, the choices can be anything for the situation. Locket is a great next item to buff up your slacking MR, and the shield is considered a must have in today's play. Dead man's plate is usually the best bet after that. For a final item, if the game is going well, sterakks gage is great. By now you have 3k+ hp, making it a great shield passive, but it also increases the damage of your sheen/base ad.

SHORT VERSION: Relic Shield -> first back sheen (this can be hard until you get used to Leona's early game strength, I usually have my adc 4+ kills by 8 minutes. If you can't get it first back, upgrade relic and get boots) -> after sheen, get sightstone/tier one boots -> finish eye of equinox/merc treads -> build and complete iceborn (glacial shroud can be bought anytime you feel comfortable)

Leona works best as either a back line diver or as a full on peel for your carry. The difficult part is deciding which of those two roles you need to fill on a game by game basis. Is the enemy team extremely mobile? Can they dive your carries easily? Best to hang back and lock down anyone attempting to snipe your Katarina or ashe in that case. Is the enemy very squishy? Do they always forget to attack your carries when you jump on their only fed player? Might be good to just continually murder that one player then.

Leona is EXTREMELY versatile and can fill nearly any role in the game. It's extremely fun to play her in lower elos where no one can stop you from killing mid and bot laners, but it is often hard to adjust to being a peeler in plat+.

NEVER be afraid to ult past level 11. The cooldown is so short by now (you should have 40% cdr by this time) that unless you full on whiff it, the use is worth it. Think you MIGHT can get a pick in that bush? Screw it, throw that bad boy down and see what happens. The slow is intense enough that even if the stun misses you will have time to run close enough for an E.