Synchro Monsters
For Synchro Summons you need a Tuner Monster, (Tuner is a subtype like "union" or "toon") and your regular monsters.
When you have a tuner monster and any number of non-tuner monsters on the field you can send them to the graveyard and add their levels (Stars) then you can special summon a Synchro Monster in your Extra Deck (previously Fusion deck) whose level is EXACTLY the same as the sum of the levels of the monsters you sent to the grave.
For example: You have a level 3 tuner
"Junk Synchron" and a classic level 5 "
Curse of Dragon" in your field. Send them both to the grave and you get a level 8 Synchro,
"Stardust Dragon" (As 3+5=8)
You could also have used a level 2 tuner, a level 4 non tuner and another level 2 non tuner for the same monster (as 2+4+2=8). But remember, only 1 Tuner per Synchro Summon
Synchro monsters will tell you if they have any other restrictions for their Synchro Summon, but most of them will just say "1 Tuner + 1 or more non Tuner Monsters" as their requirement.
Xyz Monsters (Pronounced Ik-Seez).
Xyz Monsters can be summoned using monsters with the same level, no need for tuners here.
You might notice Xyz monsters have black stars located at the opposite side of the card. That's because Xyz monsters don't have a Level, they have a Rank (They are immune to any effect that affects levels and can't be used for Synchro)
For an Xyz Summon you need to control two monster with the same level. stack them together and put an Xyz of an equal Rank from your Extra Deck on top. The cards beneath the Xyz Monster are now part of the monster and their effects will ask you to "Detach" them (Send them to the graveyard) that means, the monsters you use for the summon become ammunition for the Xyz Monster's abilities.
The cards beneath an Xyz Monster do NOT count as cards on the field
Now for an example. To summon the Rank 7 Red-Eyes above, just stack two level 7 monsters you control, and put Red Eyes" on top. To use its effect, just send the monster beneath him to the graveyard.
Just like Synchros many Xyz monsters will put some restrictions on the monsters they need to be Xyz summoned, however the most basic ones only demand 2 Level X Monsters
Pendulum Monsters
This ones are arguably the most complicated of them all, but also the most powerful, please take your time to master these guys as they are sure to become the next big thing.
As you can see, Pendulum cards have 2 effect boxes, and some red and blue arrows called "Scales"
Pendulums go in your normal deck (not the extra deck) and you can summon them as regular monsters, OR activate them as spells in the new
Pendulum Zones of the field
If you activate them as spells, they use their "spell mode" effect on the top box and remain on the Pendulum Zone until they are forcefuly removed, you can't replace them like Field Spells.
When Pendulum Monsters are destroyed, regardless of which zone they were in, they are NOT sent to the graveyard, they'll go to your extra deck. Face-Up.
To Pendulum Summon monsters, you need 2 activated Pendulum Monsters in the Pendulum Zones.You can use the Pendulum Summon only Once per turn.
First, look at the "scales" of each pendulum monster (the red/blue arrows). In a Pendulum Summon you can summon
ANY NUMBER of monsters in your hand or Pendulums in your Extra Deck as long as these 2 conditions are met.
1-Their levels are inbetween the values of the scales of the activated cards in your pendulum zones.
2- They don't have any summon restriction, like Rituals or cards that state Cannot be Special Summoned
Now for an example. If I activated a a "Scale" 1 Pendulum Monster, and another with a Scale of 10. I can pendulum summon 3 Blue-Eyes White Dragons in my hand. As their levels (8 each) are inbetween 1 and 10
this video from Konami can really help you under