In alphabetical order:
Abyssal Scepter - 2650g
+70 Ability Power +57 Magic Resistance UNIQUE Aura: Reduces the Magic Resist of nearby enemy champions by 20.
This item is good for those magic damage heroes of which the AP ratios aren't so great for, as reducing magic resistance becomes more effective than AP due to the bad ratios. Especially good on Fiddlesticks and Kennen due to the nature of their AoE ultimates and the need for magic resist.
Aegis of the Legion - 1925g
+270 Health +18 Armor +24 Magic Resist UNIQUE Aura: +12 Armor / +15 Magic Resistance / +8 Damage to nearby allied units
Great item for support tanks, and aura supports. Provides team bonuses, and the buildup for it is more useful than most. Best acquired early to mid-game.
Amplifying Tome - 435g
+20 Ability Power
Pretty basic here. On few occasions it can be a starting item, but usually not; however, it is a key item in recipes later in the game. You may be required to buy multiple of these early game.
Archangel's Staff - 2855g
+400 Mana +25 Mana Regen per 5 sec +45 Ability Power Passive: 3% of your Max Mana is converted to Ability Power. UNIQUE Passive: Each time your Champion uses an Ability their Maximum Mana will increase by 4 Mana. This effect has a 3 second cooldown. Bonus caps at 1000 mana.
A great late game item, it allows heavy casters to have a near infinite amount of mana, especially late game, as well as adding a lot of AP. Keep in mind Tear of the Goddess should not be used to make Archangel's until late into the game. More on that later.
Atma's Impaler - 2405g
+45 Armor +18% Critical Strike UNIQUE Passive: 2% of your maximum health is added to your attack damage.
A must have for heroes that need to do damage, while having a high amount of hp. Great for Olaf, Mundo, and Mordekaiser. Virtually required on Jax.
Avarice Blade - 750g
+12% Critical Strike Passive: Gain an additional 5 Gold every 10 seconds.
An good first item choice, the Avarice Blade should be rushed on heroes that need it, which include most melee DPS champions due to the fact that it can be turned into a Yomuu's Ghostblade. Heroes that need this are Master Yi, Tryndamere, and Gangplank to name a few.
B.F. Sword - 1850g
+50 Attack Damage
Your bread and butter AD item, which is used for various late game items such as the almighty Infinity Edge. This can occasionally be rushed on ranged champions such as Ashe, but is usually picked up later in the game to create much better items.
Banshee's Veil - 2715g
+375 Health +375 Mana +50 Magic Resist UNIQUE Passive: Blocks one negative spell every 45 seconds.
Great defensive item for tanks, and just about everyone else, and can supplement those that have skills or items revolving around mana, such as Manamune users. Works great on pretty much anyone that utilizes the mana aspect of it, and needs a spell shield. If your team buys multiple of these, it's one of the most annoying things to play against ever. Synergizes well with Blitzcrank.
Berserker's Grieves - 920g
+25% Attack Speed UNIQUE Passive: Enhanced Movement 2.
A situational item that should only be used for AD heroes that can close the gap quickly and have no need for other boot types. Good for Master Yi, or Tryndamere. Also note that these are the cheapest type of finished boots.
Bilgewater Cutlass - 1825g
+35 Attack Damage +15% Life Steal UNIQUE Active: Deals 150 magic damage and slows the target champion by 50% for 3 seconds (400 range). 60 second cooldown.
The active slow is very underrated, especially when ganking; however, the lack of use of the active in public games makes it's actual effectiveness not that great. To be honest, only buy this if you are building for a Hextech Gunblade.
Blasting Wand - 860g
+40 Ability Power
Core item for AP heroes, these must be stacked 2 or 3 times if you are building for AP. Useful for many of the most important AP items in the game.
Boots of Mobility - 1000g
UNIQUE Passive: Enhanced Movement 2 Passive: When out of combat for 5 seconds the effect of these boots is increased to Enhanced Movement 5.
One of the worst boots in my opinion, the only times this shoe will have benefits are for travelers like Teemo that need to get around the map, or early game gankers such as Evelynn.
Boots of Speed - 350g
|UNIQUE Passive: Enhanced Movement 1|
This a required item no matter which hero you play for obvious reasons, and it should be purchased as your 2nd or 3rd item, and in some situations, first. It builds into all the other boots.
Boots of Swiftness - 1000g
|UNIQUE Passive: Enhanced Movement 3|
Brawler's Gloves - 400g
|+8% Critical Strike|
Catalyst the Protector - 1325g
+290 Health +325 Mana Passive: Restores 300 Health and 250 Mana upon leveling up.
Chain Vest - 700g
Chalice of Harmony - 890g
|+30 Magic Resistance +7.5 Mana Regen per 5 sec UNIQUE Passive: Increases your Mana Regen by 1% per 1% Mana you are missing.|
Cloak of Agility - 880g
|+18% Critical Strike|
Cloth Armor - 300g
Dagger - 420g
|+15% Attack Speed|
Deathfire Grasp - 2610g
Doran's Blade - 475g
|+100 Health +10 Damage +3% Lifesteal|
Doran's Ring - 475g
|+100 Health +5 Mana Regen per 5 sec +15 Ability Power|
Doran's Shield - 475g
|+120 Health +10 Armor +8 Health Regeneration per 5|
Elixir of Agility - 250g
|Click to Consume: Grants 12-22% Attack Speed, based on champion level, and 8% Critical Chance for 4 minutes.|
|Click to Consume: Grants 20-40 Ability Power, based on champion level, and reduces cooldowns by 10% for 4 minutes.|
|Click to Consume: Grants 140-235 Health, based on champion level, and 10 Attack Damage for 4 minutes.|
Emblem of Valour - 800g
+17% Lifesteal UNIQUE Aura: Nearby allied Champions gain 10 Health Regen per 5 sec.
Executioner's Calling - 1350g
Faerie Charm - 180g
|+3 Mana Regen per 5 sec|
Fiendish Codex - 1245g
|+30 Ability Power +7 Mana Regeneration per 5 UNIQUE Passive: Reduces ability cooldowns by 10%.|
Force of Nature - 2175g
|+76 Magic Resist +40 Health Regen per 5 sec +8% Movement Speed UNIQUE Passive: Restores 0.35% of your champion's health every second.|
Frozen Heart - 2475g
|+99 Armor +500 Mana UNIQUE Passive: Reduces ability cooldowns by 20% and reduces the attack speed of nearby enemies by 20%.|
Frozen Mallet - 3250g
|+700 Health +20 Attack Damage UNIQUE Passive: Your attacks reduce your target's Movement Speed by 40% for 2.5 seconds (30% for ranged attacks).|
Giant's Belt - 1110g
Glacial Shroud - 1525g
|+425 Mana +45 Armor UNIQUE Passive: Reduces ability cooldowns by 15%.|
Guardian Angel - 2600g
|+68 Armor +38 Magic Resistance UNIQUE Passive: Revives your Champion upon death, restoring 750 Health and 375 Mana, this effect can only occur once every 5 minutes.|
Guinsoo's Rageblade - 2235g
|+35 Attack Damage +45 Ability Power UNIQUE Passive: On attack or spell cast, increases your Attack Speed by 4% and Ability Power by 6. Lasts 5 seconds (stacks up to 8 times).|
Haunting Guise - 1485g
|+25 Ability Power +200 Health UNIQUE Passive: +20 Magic Penetration|
Health Potion - 35g
|Click to Consume: Restores 200 Health over 20 seconds.|
Heart of Gold - 975g
|+200 Health +27 Armor Passive: Generates 5 Gold every 10 seconds.|
Hexdrinker - 1385g
|+35 Attack Damage +30 Magic Resistance UNIQUE Passive: If you would take Magic Damage which would leave you at less than 30% of your Maximum Health, you first gain a shield which absorbs 300 Magic Damage for 4 seconds. 60 second cooldown.|
Hextech Gunblade - 3190g
|+60 Attack Damage +75 Ability Power +20% Life Steal +20% Spell Vamp UNIQUE Active: Deals 300 magic damage and slows the target champion by 50% for 3 seconds (700 range). 60 second cooldown.|
Hextech Revolver - 765g
|+40 Ability Power +15% Spell Vamp|
Infinty Edge - 4080g
|+75 Attack Damage +20% Critical Strike UNIQUE Passive: Critical hits now deal 250% damage instead of 200%.|
Ioanian Boots of Lucidity - 1050g
|UNIQUE Passive: Reduces ability cooldowns by 15%. UNIQUE Passive: Enhanced Movement 2.|
Kage's Lucky Pick - 765g
|+25 Ability Power Passive: Gain an additional 5 Gold every 10 seconds.|
Kindlegem - 850g
|+200 Health UNIQUE Passive: Reduces ability cooldowns by 10%|
Last Whisper - 2290g
|+40 Attack Damage UNIQUE Passive: +40% Armor Penetration|
Leviathan - 1275g
Mejai's Soulstealer - 1235g
Sword of the Occult - 1369g
I've decided to group all these together because they are all based on the same concept, getting kills and assists to improve the stats of the item with the risk that you will lose a portion of your stacks if you die. I do not recommend these items to new players for obvious reasons, but if you are playing a high kill to death ratio champion, note that it only takes 4 stacks to make the item worth the amount you paid for it. Anything else should be icing on the cake, but don't even consider them if you don't plan on getting at least 10 stacks consistently without losing them all. They are primarily bought on tanks, mages, and DPS respectively, and all have overpowered self-buffs when you max the item out at 20 stacks which help with your type of hero the stacking item applies to.