The whole idea started when people were thinking of new ideas for the new (11-28-14 as I write this) champ, Kalista, specifically, her ult, Fate's Call. Initially, many people thought of Leona or Braum, both of which are very powerful picks for in terms of synergy. Leona has a ton of CC to follow up with, effectively keeping enemies in place for a long time in an AoE. Braum has a great ult ( Glacial Fissure) to follow Fate's Call, plus, his passive ( Concussive Blows) is absolutely perfect for Kalista's Oathsworn passive ( Sentinel's passive component), as you get to triple dip for attacking the same target.

Both of these and more are excellent ideas, they all lack one thing: forcing your enemies to fight back. With Leona, you have a very long time to put the hurt on your enemies, and Braum causes chaos in their ranks, with the ability to divide the opposition. Enter Galio. Like Maokai, Galio was originally intended to be a support. Also like Maokai, Galio is better elsewhere, but these roots are still there. This isn't like picking a support Yasuo just to capitalize on Fate's Call's knockup, there's serious potential here. Galio has a slow and strong damage in Resolute Smite, defensive and healing abilities in Bulwark, chase/escape potential for his entire team in Righteous Gust, and of course, his ultimate, which needs no introduction.

Idol of Durand is the entire reason that this guide exists. The chain ult combo potential is absurd. Fate's Call gets you into the fight with an AoE knockup, then as soon as the victims land, you have a large AoE taunt to keep them in the fight even longer, ending with even more damage. This is an amazing team fight initiate.

I've also noticed this build works very well with Ezreal, as they both have strong poke damage before going in for the kill with Idol of Durand.

Pros
Gets damage from building tanky: Runic Skin
Versatile: Scales well both as a tank and as a mage
Can demand focus from an entire enemy team

Cons
Either passive or very mana hungry
Melee range
Low harass ability: Resolute Smite is small and costs a lot of MP. Righteous Gust is a weak poke, not even worth the mana in most situations if you're only using it for damage.
Resolute Smite has a small AoE, and can be hard to land while you're learning the champ

Galio's main role as a support is a tank. As such, his AP is a luxury stat. I think of AP on support Galio like AP on Thresh: You get some of it, he scales pretty well with it, but it isn't something that you focus on.

Acceptable Marks
Greater Mark of Armor: The best marks on a tank support.
Greater Mark of Magic Penetration: You won't be using your basic attacks to harass very often, if at all, so if you opt for offense, stick with these.

Acceptable Seals
Greater Seal of Armor: Always solid choices of seals. You'll always need more armor laning a marksman.
Greater Seal of Scaling Armor: Once you're used to things with the gargoyle, these're more powerful than their flat counterparts.
Greater Seal of Health: These have become a preferable alternative to many. They're ok, but 9 armor matters more than ~75 HP later in the game.

Acceptable Glyphs
Greater Glyph of Magic Resist: These are the best glyphs for learning support Galio. They give him the early stats that you need in a duo lane, while boosting both your defense and offense.
Greater Glyph of Scaling Magic Resist: Once you're comfortable with support Galio, these're are more powerful.

Acceptable Quints
Greater Quintessence of Armor: These help boost your armor by a pretty large chunk (more than a Cloth Armor). This is useful for more than just the stats. None of the armor items are very useful on Galio aside from sheer survivability, and these allow you to breath a little longer before you absolutely must have armor.
Greater Quintessence of Magic Resist: Same story as the glpyhs.
Greater Quintessence of Ability Power: I wouldn't recommend these, but if you're going for early damage, these would be your best bet.
Greater Quintessence of Movement Speed: This allows you to be a little slower about getting boots, and they help the most of any rune end game.

Details to come.

Core Items
Sightstone is mandatory as a support. Period. You need vision, and nothing else does nearly this good of a job throughout the entire game. It also boosts your health.
Targon's Brace is a good pick for increased gold income and HP regen.
Sweeping Lens is a good replacement for your Stealth Ward once you get Sightstone. It allows you to clear wards and check objectives to deny the enemy team vision.
Mercury's Treads are the way to go on Galio, unless you plan on roaming a lot (not encouraged, Kalista loses two of her abilities by being separated from you), in which case, go for Mobility Boots. They give you MR/AP and tenacity, which is very useful as a tank.

Other Items
Abyssal Mask is the ultimate luxury item for Galio. It's dirt cheap for the stats it gives, and thanks to Runic Skin, it's even better! 2440 gold for 95 AP and 50 MR, not to mention the aura that suits your ult perfectly! Work this in whenever possible!

Spirit Visage augments your Bulwark healing and gives you CDR. Banshee's Veil stops one enemy ability, which is useful during your ult. Death Sentence and Devastating Charge are examples of abilities that ruin all of your fun during Idol of Durand.

Sunfire Aegis and Randuin's Omen are the best options for general armor.
Frozen Heart and Thornmail are the best ways to counter AS-based champs, such as Vayne. Frozen Heart is my favorite armor item for Galio, because it gives him much-needed mana, and the aura goes well with his ult.

Eventually, you'll upgrade your items: Ruby Sightstone, Mercury's Treads - Captain, Mercury's Treads - Homeguard, Face of the Mountain, Talisman of Ascension, Athene's Unholy Grail, Oracle's Lens. I suggest getting Face of the Mountain as early as possible immediately following your core, but the other items can be upgraded whenever it seems appropriate.

Max Q first. Resolute Smite is well over 75% of your damage, and each rank increases the slow by a small amount.
After that, focus on Bulwark. It just makes the most sense even from a selfish standpoint. More defense means that you can protect people (including yourself) better. If used on yourself, it also increases your damage.
Righteous Gust is really just for the increased speed up, which, don't get me wrong, is extremely nice, but doesn't benefit you or your team in the ways that your other abilities do.
Like every champ but Maokai, take a point in your ult whenever possible.