Pros:
He clears fast and safely because he has a good mix of AOE and single Target
He doesn't take much damage from jungle camps because of his W
He is relatively safe from invades since he's a strong duelist, and because he has high mobility
Due to the above, he's good at invading and counter-jungling the enemy
He's obscenely mobile with his Q and W, which means that he can initiate or escape from just about anywhere
Being energy based, he can maintain pressure longer than mana-based champions during the early game
Cons:
Being energy based, he can't maintain as much presence in long active fights. He'll eventually have to back off and recover, or just go all-in and hope that his auto-attacks are enough.
In a hectic teamfight, it's often hard for him to get to/stick on his optimal target. His kit works best when you play him as an assassin, but if you can't land your Q on a squishy target, then you aren't likely to fulfill this role very well. As a front-line bruiser, he doesn't offer as much utility as many other picks. This means that, in a teamfight, you're either the MVP by landing your Q on a carry and taking/keeping them out of the fight, or you're the benchwarmer when you miss your Q, or hit the wrong target with it, and suddenly you don't have enough damage or CC to control the enemy front line.
Early/Mid game:
This is when Lee Sin really shines. When you have low numbers of opponents to deal with at one time, you have the most room to really make your kit work for you. You have enough damage to kill most champs, and you have enough mobility to chase them relentlessly. Your shield/sustain is enough to tip trades into your favor since there will only be one person hitting it. The AS Slow from your E compounds this, and your slow is enough to stick to most targets. If you know your combos, and the capabilities of your enemies, then there is usually nothing anyone can do against you except hug a tower, call for backup, or blow all of their spells/summoners defensively (or all-in you, in which case you disengage immediately and make them waste whatever they just used to try to kill you).
Late game:
As I noted in the Cons, late-game can be a bit rough for Lee Sin. The shield that got you through early and mid game suddenly isn't enough against the force of their entire team. Your attempts to jump on squishy targets are prone to being suicidal, and your damage is nothing special against anyone who build any defensive items whatsoever.
However, Lee Sin does control objectives pretty well. His Q and his passive are really good at securing/stealing Dragon and Baron, and he threatens a pretty potent dive when you have an enemy turret under siege. If you save your R for a disengage in case the enemy initiates on your team (most notably, your carries), throw out Qs fishing for a hit on their own carries, and use your W to shield your ADC from enemy poke, then you can provide a huge amount of security for your team as they chip away an enemy turret. Just remember that the tower and inhibitor is the objective, not kills. Don't land a random Q and jump to your death without a very good reason, and don't waste too much energy trying to be fancy by jumping in and then back out again with your W- a smart enemy will note that half of your abilities are on CD, and that you're nearly out of energy, and take that opportunity to initiate, and you won't be able to respond.
For items, it depends on your own team comp, and the enemy team comp. From the jungle, a more tanky/bruiser style is generally easier since you'll have lower income than a solo lane. You want just enough damage to be a threat to their squishiest members, and then durability from there. If your team is lacking damage, or if someone else is covering the tank/bruiser role, or if everyone on the enemy team is squishy, then you could consider going full damage, but just remember the squishier the enemy team is, the more damage they generally do.
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