There are many reasons for jungling with Poppy, and why it is not more common i do not know.
One of the reason i look at her as a good jungler is the fact that her Q ability, Devastating Blow, Deals more damage the more max health the target has.
"Poppy enhances her next attack so it will also deal extra magic damage and 8% of the target's max health as magic damage. The magic damage based on 8% of the target's max health cannot exceed a threshold based on rank."
This will help her take down creeps such as Dragon and Lizard very fast.
But also her abilty to gank is very good indeed. Her W ability, Paragon of Demacia, grant her increased movement speed, and with ghost aswell you can chase down any enemy champion that is fleeing. Also the E ability, Heroic Charge is a great distance closer when ganking, and with good placement you can stun your target for 1.5 seconds.

The item you start off with is the Regrowth Pendant, this item is great for many reasons. While in combat you regenerate lots of health. And when you for example have just finished the wolfes at level 2 your are really low on HP, but you have to run all the way to the small golems afterwards soo by the time you reach the golems you have more then 400hp again.

Start off with the Wraiths, smite the big one and finish off the rest, afterwards go and kill the wolfes, after the wolfes you go and get the small golems, use your smite on the last one.
After barely finishing the golems teleport back and buy Philosoperstone. You may then go and get blue buff, go and heal before u go and get red. after healing, if your quick enough you may have both buffs and may then go on your first gank. After playing this build 7 times i have still not once managed to fail on not getting a kill or assist on this first gank. After this you should just get the other teams blue buff and then go to a lane.

After you have done your first gank and gotten the blue buff again you should go to a lane(top/bottom) which needs your help at that moment, or even if none needs help you should go and push a tower. If you get a tower or a kill while laning you should leave the lane and do some jungling, for example get red, blue and wraiths, before you go to the other lane (bottom/top) and try to help them do the same thing. And unless there is building up to a team fight middle you should just try to get the tower at the lane you are at.

At the mid-game stage you should, with some decent minion-farming and a couple of assist or kills you should have finished you Mercury Treads and your Trinity Force. These are what you may call your core items. They ensure you great movement speed (the trinity force grants you 12% movement speed) which helps you chase down enemies even easier then you usually would with the W ability, Paragon of Demacia, and ghost. After this you should just focus on getting a Hextech gunblade. The Q ability, Devastating Blow, deals more damage if you have AP and AD, this makes Hextech gunblade the ideal item, since it as well gives you spellvamp and lifesteal. The spellvamp and lifesteal helps you stay alive for way longer in a team fight. At last you should get an Rabadons deathcap just for the massive ap you get. Sometimes at very long games i do not bother upgrading the Philosoperstone i just sell it and buy a Frozen Mallet. But it is very rare that a game lasts that long

Its pretty simple how to choose your target for the ultimate and who to attack in a team fight, but it is VERY important that you do not just attack a random meelee champion. you may very well turn the tide of a battle if you manage to quickly throw your R abilty, Diplomatic Immunity, at their tank (for example Shen, Cho Gath, Amumu and similar champs) and then quickly run behinf them and go for their squishy AP champions, Since they cant deal damage to you. You then have 6 seconds of pure joy ******* up their main damage source (the casters)

Poppy is great in laning aswell, and this build works very well for laning aswell as jungling except the smite. She is totally awesom elaning with another stunner. I have personally played a lot with a Renekton. In the early stages of the game this is totally freaking insane, if Renekton gets in a good stun in the brushes you can charge into the brush and the target will be dead before your stun is over. Acctually often use a flash, ignite combo. This makes it so that as soon as Renekton gets the stun i flash out of the brush and charge in. I then throw ignite at them and give them a smack with my Q, Devastatin Blow, and they are usually dead.

Other items that are extremely good for Poppy is Phantom Dancer. Phantom Dancer gives you the very vital movement speed that you need for chasing down you kills. Gunisoos Rageblade is also an amazing item for Poppy, Since her Q ability, Devastating Blow, which builds on both Attack damage and Ability power. The fact that Gunisoos Rageblade increase her attack speed is also very good since i build her with quite a high amount of Attack damage (the hextech gunblade).

Poppy is a very good tanking character even though i have barely tried it I can say with good consiounse that she is very good for tanking. She can stun and dodge alot of damage by using her ultimate. But the most important reason i consider her a great potential tank is her passive.
Valiant Fighter: Any damage dealt to Poppy that exceeds 10% of her current health is reduced by 50%. This does not reduce damage from structures.
This means that if a Tryndamere which crits 1000 starts smashing you, you will only recive mere 500 damage..