There's really no thought behind how this build works. I find it works best on Morgana and Anivia, but it works for any AP-able champion (even unorthodox ones like Cho'Gath, or Kog'Maw) who uses mana.

You rush through the items, however you may be able to, and you gradually scale through the game in every important stat.

Endgame, this gives any AP Champion an immense pool of mana, nearly 4k HP or more, and a minimum of 700 AP or so. With all the Rod of Ages completely full, max possible mana, all elixers and a Baron Buff, you're rocking an easy 950+ AP, while being (armor/MR squishy.. but: ) rather Healthy. With so much mana, the only time you should need to recall is after a teamfight. The worries are thrown out the table. You can take a blue, but should probably leave it for the not-so-tanky carries or tanks.

You may be focused, but the enemies will exhaust a lot of abilities on you alone, leaving great opportunities for your team (hopefully, if not one-sided fighting) to isolate and decimate your foes.

Of course, if you're NOT focused, you will be in a great enough opportunity (due to cooldowns from the boots/elixers/possible blue + masteries) to deal an immense amount of damage to anyone near you. The only struggle is with a tanky opponent, or someone with a large amount of sustain that can plow through you. However, if there are such opponents, your team should be able to compensate.

It's an easy enough followthrough, and doesn't take much effort. Just do your best to farm, and rock those fights.

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Only note: This is on the line between AP Carry and AP Offtank, so interpret is as you wish.

Most AP Carries have an easy time farming with AoE abilities, so this shouldn't be much of a problem early game. The struggle is finishing that Second RoA and getting your first 300~450 AP. That's about midgame when things get crazy.

If you CAN AFFORD to ignore your team for a few teamfights and farm/push a lane, or farm the jungle (preferably farm/push a lane), by all means. DO NOT do this if your team may or may not absolutely need you. If you can't risk it, don't do it. It's better to get Kills and Assists than farming, as far as precedence goes.

If your team doesn't particularly need you, farming minions may be more worthwhile, or even in value, than participating for the assist pool or getting those kills. It all depends on situation.

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If you're having trouble farming, you may need to begin roaming to other lanes your Jungler (if present) is not around. If you're mid (primarily should be), and your jungler is actually jungling, ask for help to zone out your opponent. If it's a stalemate M+J 2v2 in mid, just wait it out and poke the lasthits down. If you can't get any, powerhouse the minions down at the expense of some mana. You'd probably be in a position to recall anyway, at that point.

If your jungler is bottom, and you're pushing past mid and have an mia enemy mid, by all means go up top, even if they're pushing the tower, 1v2 or not. By the time a mid usually gets there, an enemy will have appeared, or you can help push down a turret. Either way, it's an ample opportunity to score at least an extra 20 gold before your mid returns and pushes it back, OR the enemy mid tower had pushed it back. [This scenario is before any mid towers had gone, though. You may have to interpret it as the situation escalates.]

The biggest problem is if you're playing as a champion with a hard enough time to farm in the 1v1 situations as it is.

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Annie, Fiddlesticks, Fizz, Kassadin, Karma, Syndra, Sona, Zyra, Veigar, Nautilus, Nidalee, LeBlanc, Heimerdinger; to name all that come to mind, that may need to focus on poking the enemy, and/or conserving mana for an attempt at a kill, more than farming creeps all lane.
This is simply my opinion, with a second opinion judgement. Your opinions may vary.
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If you're having a hard time farming because you're being ganked constantly, under your turret all the time, or simply cannot safely farm without being forced out of lane, you may need a major "babysitter," even if that means your jungler saps some Exp and loses CS, or one top may need to assist you in mid frequently. With such action, however, note that the enemies will be aware of what's going on and may react accordingly.

(Although not explicitly stated elsewhere, this will work for supports, if they get enough assists. Not recommended, but viable.)

To name a few:

WILL WORK:

• Ahri
• Amumu
• Anivia [Works Third Best]
• Annie
• Blitzcrank
• Brand
• Cassiopeia
• Cho'Gath
• Elise
• Evelynn
• Fiddlesticks
• Fizz
• Gragas
• Heimerdinger
• Karma
• Karthus
• Kassadin
• Kog'Maw
• LeBlanc
• Lulu
• Lux
• Malzahar
• Maokai
• Miss Fortune
• Morgana [Works Second Best]
• Nami
• Nautilus
• Nidalee
• Nunu
• Orianna
• Ryze [Works Best]
• Sejuani
• Shaco
• Singed
• Sona
• Soraka
• Swain
• Syndra
• Twisted Fate
• Veigar
• Viktor
• Xerath
• Ziggs
• Zilean
• Zyra

WILL NOT WORK:

• Alistar -- Needs more support viability, or straight tankiness, that this does not supply.
• Diana -- Requires more armor or magic resist in common situations than this build will supply.
• Ezreal -- Focuses too much on the health and mana pool to build into AP, where AP Ezreal will require a much more steady AP gain to be majorly useful in most games.
• Galio -- Needs much more Magic Resist and/or Armor towards his passives, mostly for his ultimate.
• Janna -- Should not be played AP, if played at all. She seems pretty dead these days.
• Katarina/Kennen/Mordekaiser/Rumble/Vladimir -- All don't use mana.

The only example I can give you is this:

League of Ra - Anivia Legendary Skin Testing

Features the entire (final) build, and what it can do, but more in particular to showcase the stats properly.