This build is one I thought up only recently, around Easter 2012. It takes Leona's strong CC and mobility and allows her to be played as a strong, tanky jungler. This build must not be taken as the be-all and end-all though, as this build is how I personally like to play her. Others are fully entitled to disagree.
My rune page is set up to give me a good amount of survivability and strength in the early game, as later on her items have a far more profound effect. The attack damage marks are there to slightly increase my base DPS and help me clear the jungle faster. The armour seals and magic resist glyphs are there for your standard tanking loudout.
My masteries are 2/19/9, putting as much into the defensive tree as possible. This allows me to still fully accomplish my role as my team's tank, to jungle safely, and to allow a good amount of mobility thanks to the Initiator mastery. The 2 points in offence are to further increase my damage against the jungle mobs. The utility points are mainly there for some more movement speed, some extra mana, and the crucial buff duration mastery. In my mind, this is probably the best setup for a jungling Leona, as mobility and durability are her two key aims.
My item build is pretty typical of a tanky jungler, with the exception of the use of Wriggle's Lantern. I take it because I am a fan of the lifesteal and vision I can get from it, but I accept that others may not want to use it. You could modify this build by putting the cloth armour instead into an Aegis of the Legion, or remove it in favour of a Regrowth Pendant, which could then be built into a Shurelya's Reverie or Eleisa's Miracle. It usually does not make too much of a difference, but I find that, in the late game, exchanging Wriggle's for an Atma's Impaler makes you even more tanky and capitalises on the health that has been stacked up thanks to Tri-Force, Sunfire and Rylai's. Potentially, using Unholy Grail could be helpful, allowing you to use your CC's more and deal more damage.
This skill sequence, in my mind, is probably the best for playing a mobile, tanky Leona. Maxing Eclipse first is a must for its AoE damage component. I guess it can be debated which of the others to level first, but I level Zenith Blade first to reduce its cooldown and thus allow me to jump around teamfights. Also, doing it this way is more mana-efficient, as Shield of Daybreak's mana cost scales with level.
Standard for a jungler. Since Leona has no 'free flash' from her abilities, she requires Flash. Smite is also a must, as at low-level you will struggle on the buffs.
As I stated in the introduction, this build is how I personally like to play Leona, and so I recommend that you check other Leona builds to make sure you find the build that is best for your playstyle.
If you are going to use this build for ranked games, be very wary in the time before you have your Wriggle's Lantern, as you are very vulnerable to jungle ganks. Getting into the mid game, you need to be ganking as much as possible, as Leona's strong CC should get you a lot of kills and assists. Also remember that you are the team's tank, so you should not be getting kills too much. My personal best K/D/A is 7/1/16.
In teamfights, remember to stay at the front of your team, and try to tank as much of the enemies' damage and CCs. Try to use Solar Flare to stun as many of the enemies as possible, but try to target their AD carry specifically. Use your abilities as much as possible to maximise your team's DPS through your Sunlight stacks.
Creeping / Jungling
When jungling, start at the blue camp. Get your mid player to pull the golem and stick around for some extra damage. Finish it off with Smite, and the do the wolf camp. Do the wraith camp next, but do not get red at this point. Take out the golems to get you to level 3, then look for ganks. After this early ganking phase, go back into the jungle, take red and continue clearing out your mobs. Gank whenever you see the opportunity.