Galio is a diverse movement speed manipulation champion. His speed with this build will allow him to be there for the fights or ganks when needed. Galio is a gank focus jungler, he is excellent for ccing with his slow Resolute Smite and allowing his allies to catch up with his Righteous Gust and if need be initiating and surviving with his (Q,E,W,R) combo. This build in particular will allow you to kite enemies with great difference or initiate when needed.

At the time I'm writing this Galio has a 51% win rate counting 29k games according to na.op.gg making him one of the least played champs in league. Though you should alwasys consider what you're up against when picking a champ, Galio's viability as a tank or apc jungle makes him a good pick against the right teams. Basically If the enemy team is AP heavy or if your team is AD heavy, missing a tank, needs a peel, or initiate Galio would be an excellent pick. Of course there are exceptions, champions you shouldn't be picking Galio into, you can view them in "threat level" as well as the explanation in the chapter of the same name.

What to avoid?

Anyone who can cancel your ultimate with an augmented aa atk ( Garen, Blitzcrank, Xin Zhao, ect) should be accounted for when planning to initiate, wait for the augmentet attack to be burnt before you initiate. This highly effects consideration for threat levels.

What can't I take on?

Galio is good a closing the gap, making him great for catching burst and poke champions. Sustain AD damage and tankier champs are big problem and shouldn't be fought without the help of your team. Avoid soling champs like Poppy, Udyr, Darius.

Scaling Magic resist: Though Glyphs of magic resists are viable and do help deal slightly more damage early on, Galio is more of a late game champ so I've scaled his MR for such.
Armor: Inversely seals of scaling armor would be nice for late game, but in order to sustain in the jungle while bulkwork is on cd you'll need flat armor
Magic Penetration: Magic Pen Marks are really the only option here offering more viable damage than any other mark early and late game
Movement Speed: Movement speed is essential for this build, these quints will help with early ganks and late game initiate as well as allowing you to dodge every possible skill shot

Heres my explanation for the masteries, starting from the right

Resolve

Unyielding : Armor helps for jungling as well as the MR for passive's damage
Explorer : A staple for high mobility junglers, 15% movement speed helps, a lot
Runic Armor Stacked with Bulwark and if needed Spirit Visage will make Galio a true off tank
Insight The cooldowns will help a bit, but its far more useful than Perseverance because bulwark's heal isn't base hp regen


Cunning

Wanderer movement speed out of combat helps more than single target damage that he doesn't have much of
Runic Affinity As a jungler he'll need longer buff durations, plus Secret Stash doesn't effect Hunter's Potion
Meditation Galio sucks up so much mana for prolonged jungling or fights he'll need the mana regen
Dangerous Game Thought not particularly needed, still more useful than a support's mastery >.<
Intelligence So many viable items on Galio give CDR which is a good thing because he'll need it to effectively sustain his manipulated movement speed. Breaking the cooldown cap allows for a slow lock down :D
Thunderlord's Decree Allows Galio to deal some extra damage for your QWER burst combo, although Windspeaker's Blessing would be awesome, it doesn't effects Galio's Bulwark