Pick a champion with a fairly healthy three camp clear.

Pick a jungle champion with some crowd control.

Pick a jungler with gap closer.

Pick a jungler with some damage.

Pick a jungler with decent jungle camp clearing speed.

A good jungler combines the following traits: Good sustain. Crowd Control. Gap closer. Good damage. If they have all of these traits they can gank practically any lane and more importantly they have options as to how they want to influence the map.

Gank
Countergank
Farm
Counterjungle

If your jungler lacks the sustain to get through three camps with a decent amount of health all of the other options are pretty much off the table. They have to back and buy. Which may work out fine if your laners are holding their own and the enemy jungler hasn't already shown up to the tip the balance of power in the lane. But why be reactive when you can be proactive? Or to put it another way if you can you want to control the tempo of the game. For instance you can be 90% sure that if you gank all three lanes and get kills in all three lanes you can bet that the enemy laners will start to apply pressure to their jungler to start ganking even if it is someone like Master Yi that has fairly meh ganks prior to Level 6 when he is still only going to be able to bring damage to the equation since he lacks any form of crowd control. Ganking really isn't Master Yi's win condition. Farming to 6 and getting his Enchantment: Devourer stacked is Master Yi's main win condition.

Amumu's win condition is a combination of ganking, farming, and altering the course of team fights with his monster ulitmate.

Xin Zhao's win condition is usually ganking early and often to start snowballing, which is why I cringe when my laners play super aggressive when there is a Xin Zhao jungler on the enemy team.

Shaco's win condition varies. He can look to make the enemy jungler useless or gank to get his laner's ahead or a bit of both. Usually it is both because he generally wants to finish out the game as fast as possible because his team fighting is meh unless he can instantly assassinate one or more of the enemy team's carries.

Why is Gragas back in the meta after having a dismal 44% win rate in soloqueue? He has decent sustain with his passive Happy Hour and the damage reduction he gets from Drunken Rage, he has good CC in the form of a slow from Barrel Roll, and a stun and gap closer in Body Slam.

But after a number of nerfs and tweaks to his barrel damage he just couldn't clear the jungle fast enough compared to the current crop of top tier junglers. The tweak to Drunken Rage that made his post-swig autoattack do a small amount of AoE damage gave him just enough umpf to get out of the rut that Riot had dumped him into after a long time at the top. Now he is much harder to bully in the early jungler and he can keep pace with the enemy jungler in terms of items and ganking.

Graggie is an ideal jungler based on what I just discussed. On top of that he can Body Slam through many walls and he can use Explosive Cask to engage or disengage. There are very few champions that boast a similar level of utility while bringing a heap of tank and a side of damage.

So the point of this is not really to be a Gragas fan boy. It was really to illustrate the characteristics that make a top tier jungler top tier.

You can pick your Master Yis or Tryndameres when your team already has a healthy amount of CC and/or tankiness and they need more damage. Their CC will make your ganks much more reliable than they are just from your kit alone.