Let's see our friend the wiki! "Hybrid refers to using two or more champion build types. These combinations can be beneficial, and can allow a champion to be more diverse in terms of its combat effectiveness."
The idea is simple. More versatility = better champion. Of course, it's must be BENEFICIAL to your champion. For instance, I don't want to see a Karthus building AD. That's doesn't benefits his panel of skills AT ALL.
There is many ways to mix build to add versatility. What are those ways?

BLUE BUILD
What is actually a blue build? Simple, it's a AD build mix to get benefit's from

PURPLE BUILD

YELLOW BUILD

RAIN-BOW

Mark
Mark of Hybrid Penetration When your champion is 50/50 damage of pyisical and magic like Jax. Take those.
mark of armor penetration Your character is mainly physical oriented like Tristana, take those.
Mark of Magic Penetration Your character is mainly magic oriented like Katarina, take those.

Seal
There is no specifical rune for going hybrid.


Glyph
Glyph of Magic Penetration The only good choice if you want to boost the damage of an magic oriented champion like Katarina. Hybrid benefit's both from AD and AP for boosting their spell, but their source of damage still be mainly magical like Katarina.


Greater quint
3x Greater Quintessence of Magic Penetration Like said before
3x greater quintessence of armor penetration Like said before
3x greater quintessence of hybrid penetration Like said before
1x Greater Quintessence of Life Steal/2x Greater Quintessence of Spell Vamp Those runes with the mastery Vampirism and the Hextech Gunblade and you will turn into a sustain beast! Two spell vamp because healing by burst spell damage is better than auto-attacking.

In this section I talk about specific mastery which are the most useful when playing Hybrid.

Damage tree
Those tree masteries are the best you can take. All the other mastery boost your AP or your AD. Only 17 in this tree can be sufficient.
Weapon Expertise Percentage armor reduction > Attack Damage
Arcane Knowledge Percentage magic resistance reduction > Ability power
Havoc This make you hurt, with both your physical and magical abilities.
Those two masteries is the best when you want to go deeper in this tree.
Sunder This is better than AD, so for pure damage dealing it's good.
Executioner The only reason you go deeper in this tree

Defense tree
There is no difference when choosing masterie in the defense tree when playing hybrid than when playing pure AD or pure AP. But I love to go 17/13/0 for the best mix of damage and defences.

Utility tree
Vampirism This bad boy gonna give you some sustain. Since you are hybrid, you benefit BOTH the lifesteal and spell vamp. When jungling it can be a nice combo with Lifesteal or Spellvamp Quint rune.
Nimble Speed is awesome, but if you go this much far in this tree it's because you have wanted to take the next mastery.
Intelligence This masterie can be really good in combo with Sorcery for a sweet 10% cooldown reduction.

For hurting anytime anyone
Manamune/ Archangel's Staff
This small sets is unusual and can be quite effective. It cost a total of 4800 gold only. Alone they are strong, together they become even stronger.
Champion have between 700 and 1400 mana, so let's say 1000 base mana +2000 mana with those item for the math.
-The Muramana will give you 180AD when toggle and 60AD with the passive. Total melee AD from it = 240AD. For Blade of the Ruined King to hit 240AD, the target need to have 4300 health. Wait, Muramana actually cost less gold? Over powered!
-The Seraph's Embrace will give you an 750 life shield with it's active and 90AP with the passive for a total of 150AP. It's like having a mix of the Rabadon's Deathcap passive and the Zhonya's Hourglass active. Wait, Seraph's staff cost less gold? Over powered!
-Just make sure than your hybrid really benefit's from playing both like a mage and like a carry. For instance Twisted Fate and Ezreal both benefit's from casting spell in a short combo or just spamming attack on one target. But in the contrary, Kayle doesn't benefit's this. He's strong with auto attack, so boosting skills AND auto attack is a waste because boosting ONLY auto-attack will benefit's him more.

Vs armour/magic resist stack
Last Whisper/ Void Staff
We all know than building Armor penetration or Magic resist penetration is actually better than building pure AD or pure AP.
We all know than building Percentage penetration is better than Flat penetration.
These two bad boys mixed with the Weapon Expertise / Arcane Knowledge masteries will give you a good 43% armor/magic percentage penetration.
Is there any interest into building any armor or magic resist against this combo? Not that much... It's far better than building other Flat penetration in term of gold efficiency than any other item.

For focus targeting
Black Cleaver/ Wit's End
You want to rock and make lot of damage? You want than your team does too?
Take these two items and you will feel like Kayle with his Holy Fervor and you will make your teamate deal greatly higher damage. Those are good pick for fighter has it give defence and damage boost, like Kayle. Those items are bad for squishies champions. You take those because you want to hurt and because you can take more hit.

For good benefits from ganking
Sword of the Occult/ Mejai's Soulstealer
You want to rock the sooner than possible? You want more to spend time killing your enemies than killing creeps? Take those.
This combination is quite effective on champion like Evelynn or Katarina. The gold efficiency of those item make killing enemy champion more usefull than creep for cumulate the more stack possible. This item combination is for player who have GREATZ skills.
Everybody will want to make you loose your stack, so be sure to have an escape tool. For example katarina Shunpo. Or a way to avoid unwanted fight, like Evelynn Shadow Walk.

Versus health stack
Blade of the Ruined King/ Liandry's Torment
The opposite team is massively staking health? Make them regret it.
Those two items are quite good. Those gonna make you hit hard with spell's and auto-attack. Big down side. They scale with the ACTUAl life of your opponent. The lower in life they goes, the harder it his to kill them because you will hit less hard. It's mainly usefull if the other team have strong tank or play really tanky.
Let's say than the other team has Dr. Mundo. He will have about 4500 life sometimes, which will be a real pain to kill. With those item, your auto-attack will deal 225 bonus damage and your spell will deal 405 bonus damage (if your target is slowed 50% of the time during the three second passive).

To boost your team mate sustain
zeke's herald/ Will of the Ancients
You are a support and you benefit's both AD and AP? (like Alistar or Blitzcrank) You want to boost the sustain of all your team? You want to boost their damage a litle too?
THE AURA BEST, YOU WILL BE! Whenever you kill an enemy, 10 to 20% of his life will be given back to your team. The AD and Ap boost is nice, but the real nice boost is the life steal and the spell vamp. One of you allies need heal? Follow him in jungle or an other lane to let heal himself on creep with your aura!

Classical hybrid core (auto-attack oriented)
Guinsoo's Rageblade/ Hextech Gunblade
Champion like Kayle or Varus

Classical hybrid core [abilitie oriented]
Hextech Gunblade/ Trinity Force
Champion like Evelynn or Katarina

In this section I show you which character is actually a hybrid.
There is the actual Classical section which talk about champion who are solid pick when played hybrid.
There is the actual Unclassical section which talk about champion who can be good when played hybrid, but have not been played that way that much.
There is the actual Possible section which talk about champion who haven't intended to be played hybrid. This section is not for poeple who can't manage to play outside classical meta.

Classical hybrid
60/70 Kayle
70/70 Jax
50/80 Akali
60/50 Alistar
80/60 Shaco
60/50 Poppy
40/90 Katarina
60/60 Twisted Fate
70/60 Ezreal
70/50 Irelia

Unclassical hybrid
90/50 Tristana
50/70 Sion
40/70 Evelynn
80/50 Kog'Maw
80/40 Udyr
80/30 Lucian
80/30 Xin Zhao
60/60 Yorick
80/50 Miss Fortune
80/60 Corki
70/60 Kennen

Possible hybrid
60/70 Nasus
40/90 Orianna
50/70 Gragas
70/40 Volibear
80/40 Hecarim
70/40 Varus
70/80 Diana
60/70 Elise
50/70 Nidalee
60/70 Fizz
50/70 Teemo
50/60 Thresh
40/50 Taric
40/50 Blitzcrank
70/50 Skarner
40/80 Zyra
30/80 Kassadin
30/80 Rumble
30/70 Cho'Gath