League of Legends (LoL) Question: I'M SO CONFUSED WITH ALL THE AP ITEMS!

So I'll leave a tl;dr here because your question is too broad for a Q&A thread. You should do a thread in the corresponding part of the forum for higher chances of in-depth answers. Don't be too afraid of misplacing it within the forum, an admin will move it to the correct subforum if you do (but don't abuse the good will of the admins and try to learn the correct spot for stuff over time).

First of all, the "raw stats" items: Morellonomicon, Rod of Ages and Archangel's Staff. These give almost purely stats (AP/HP/mana/CDR) as opposed to more utility based items, which tend to give some impactful active/passive.

- Morellonomicon gives decent AP/mana, 20% CDR and a nice passive against enemy healing. It's a great starting item for a quick boost of stats, as opposed to more late-game focused items like Rod of Ages and Archangel's Staff. It's a must buy against heavy healing champions, and can be bought mid/late game for the stats if you need the 20% CDR (otherwise buy some other 100 AP item).

- Rod of Ages should be your first or second full item, third one at most, since it takes 10 minutes to charge. Buying this late in the game could mean having a subpar item. You should buy this when you don't want your powerspike "right now". It also adds some survivability in the form of HP but it isn't too much anyways.

- Tear of the Goddess into Archangel's Staff is the de facto late game raw stats item, since it'll give you more AP than any other item (except maybe Rabadon's Deathcap) but it needs to be charged and it can take a long time in some champions. Notice that it has great synergy with Rod of Ages. Some late game champions buy tear into rod and then finish the archangel's. It's not compulsory to buy both though, as it means a lot of time not being too much of a threat.

The rest of the items have some interesting active/passive that should be the reason for buying it. The itemization landscape is ever changing, but everyone of these follow a pattern.

- Athene's Unholy Grail is support oriented, so it's only useful for AP supports or for some AP mid laners if they have a shield to proc the passive and even then it might only be bought if they're behind.

- Hextech Gunblade is mostly balanced for hybrid damage dealers, mainly for Akali. It might sound good in theory for many champions but trust me, it sucks on most if not all of the AP builds.

- Hextech Rocketbelt is mostly an initiation extender (see: Annie Flash+this active+ Summon: Tibbers stun). It gives less useful stats than the rest because the active is very powerful. You should buy it only if you plan on abusing the active in some way.

- Everfrost is just a filler item. It has a ranged slow in his active and gives decent-ish stats. Time will say if this is useful and who would want it. It seems useful in some mages (i.e. not AP assassins) and maybe supports. It also seems like it is underpowered.

Hope this helped, and hope it didn't turn out to be too long :P
PS. If you're having doubts about what to build you should look in pages like probuilds to see where they're being built by the pro's. Sometimes they do it for specific reasons you might not be aware of, so be mindful on that and try to see where every item fits over time.

1. Morellonomicon
a. But this if you need mana, damage and CDR to be useful with your kit (it's a good first item), if you are out to get kills and assists frequently then this item allows for mana to be regained by doing so. It's also good if the enemy relies on sustain and you want to counter that, so if you are poking your opponent a lot in lane then this can be great as it reduces healing and regeneration whilst under 35% by 40% for 8 seconds which is great!
b. Cooldown, Mana, Damage and Healing Reduction. Good lane power spike but doesn't offer sustain unless you have a built in heal in your abilities.
c. Get it if you need mana in lane, as it builds out of Lost Chapter which is fantastic for mana early on, also get it if you need to prevent high sustain champs from out-lasting you.
d. High poke champions can do well with this since the stats are fantastic, build I'd recommend it on some who you are confident you can strive for kills on, so LeBlanc, Morgana, Ziggs, Brand and various other people who can get kills in lane quite effectively.

2. Athene's Unholy Grail
a. It offers both mana and health regeneration in lane whilst also giving an abundance of various stats that compliment champions who heal allies in fights.
b. This offers great sustain for allies whilst also giving you lots of utility via stats on the item itself, with the health regen, mana regen and magic resist you can fare better in longer fights and also will have a good balance of stats.
c. Pick this item if you have a healing effect that affects allies, this is considered a support item now due to it's effect only working with allies so getting it on champions that take damage in lane, require mana and can also utilize cooldown reduction to get an advantage in long or short fights.
d. Janna, Soraka, Nami, Bard, Karma & Sona, naturally you won't always want this item but it can be really useful if your need to provide more healing then usual and are at risk of running low on health / mana yourself in the meantime.

3. Rod of Ages
a. This is an extremely strong item now due to the abundance of stats for a cheap prince, additionally thanks to it's scaling nature it makes for a solid early game item as your midgame will become incredible. It also offers health generation and mana generation through burning through them respectively.
b. This item gives great sustain for both health and mana, a large pool of both stats and a good amount of AP for it's price. It scales to give around 66% more stats after 10 minutes making it very cost effective.
c. Buy this against high burst opponents as the increased life may give you enough life to survive the burst, also if you're playing someone who is susceptible to poke and can also use their mana quickly then this item is great as the sustain via the respective stats is quite valuable especially against poke. Try to get this item early as the scaling needs to be made use of to get full effect.
d. People with high mana costs, lots of poke and partially ap-bruiser nature can make use of this quite a lot, especially if getting in the fight is important, so [Kassadin]], Gragas, Swain, Anivia, Ryze and occasionally Singed.

4. Everfrost
a. Look at Rod of Ages, remember the 400 gold increase and the slightly higher base stats on purchase, but you also get a slight damage utility increase through the active, it does moderate damage and applies a minor slow.
b. At slightly more AP and Mana then Rod of Ages you have a good early-advantage item, especially with the active that provides a minor slow and slight damage which can add to your burst.
c. Get this item if you need to proc a spellshield ( Banshee's Veil & Sivir have one), need to get a early lane advantage (first-item really early lead), or need a minor slow to make better use of your kit (not sure, but can help land skill-shots), it's also good to keep melee champs at bay if they attempt to get to you.
d. Usually the same as Rod of Ages, except Veigar can also be seen grabbing it to apply slows so he can better land his stun (and it's damage scales well with his passive).

5. Hextech Gunblade
a. It gives sustain from both AP and AD abilities, on-top of that it gives a bonus damaging effect that applies a slow which can allow you to start close to your target, finally it gives AP & AD which means hybrid champions get a good amount from this item.
b. It's slow is extremely strong at 40% for 2 seconds, this is higher then that of a lot of other items (including the Everfrost which has a slightly higher % at 65% but for 1/4 of the time at .5 seconds). A mix of AP and AD and sustain from ALL attacks, which can't be found on any other item.
c. Take this item on champions who can make full use of AP and AD, especially if sustain is welcome. Build it against people who attempt to escape or create a distance between you and them as the slow counters and prevents this.
d. Amazing and usually core on Akali, it can also be taken but not as frequently on Kennen, Jax and Diana. Anyone that needs sustain and can make use of the slow to land more attacks / stay close to their opponent.

6. Hextech Rocketbelt
a. It gives you a gap-closer, 1/4 of your max CDR, 60 AP which is not a bad number for an item with built in utility, 300 Health which is actually quite useful for AP champions (especially ones who build / have resistances in their kit already) and it gives a slight damage effect with the dash that can be used to finish off opponents or help add to your burst on demand.
b. A good mix of stats for a first item (no mana / regen is can be quite bad for some champions) and a utility gap-closer which is great for getting close to opponents to land attacks.
c. Take this whenever you feel their range leaves you at a disadvantage, or if you need some health to survive burst, but also need some burst yourself (though Everfrost could do this also), also if you want to catch your opponent off-guard then you can use this to break the safe distance they create and possibly force a flash.
d. This item is fantastic on champs like Annie as they need to close the gap to really dominate, it can also work on champions like Lissandra, Swain and a lot of AP champions if you have a Taric on your team (he links with you, you dash close to your enemy and he applies stun), it can also work well on Galio to position his ultimate better quickly in a fight.

7. Tear of the Goddess/ Archangel's Staff
a. Buy this because it gives insane mana which works well if your champion burns through mana quickly, benefits from a high mana pool (see Ryze / Singed), has high base damage from spells and just needs the ability to use them frequently or if you want to build into the final item for it's utility or damage increase.
b. A lot of mana for a cheap price. Can work on anyone that mana can be an issue on.
c. Take this if you run out of mana frequently, it gives so much that often times you'll never have mana issues after buying it. As for Archangel's I'd say get this if the shield will function well as effective health or if you have other mana items to make full use out of the mana to AP scaling (See Ryze agian). If you need to spam spells to be effective in lane then you could just grab a tear and work on some ap items, even if you have no intention on upgrading to Archangel's it could give you the edge in lane to win the game regardless.
d. Ryze. 1000x Ryze. Swain can also get it along with Singed, Anivia and Ezreal since they use a LOT of mana and can make full use of this item. Though you can buy it on anyone where mana becomes a problem if you intend on throwing out many of your abilities. Archangel's is trickier but I'd say anyone that builds enough mana ontop of tear to make full use of the bonus AP (especially champs that get in the fray so they can use the shield).

I hope I covered everything, I may have missed a few things and made a few mistakes but I hope this is helpful regardless!