The side deck is for rematches. You can rematch your opponent, but you cannot change decks. Think of it in a tournament sense. Matches are I believe 2/3, and a side deck provides you the opportunity to switch in cards or swap out cards depending on your opponents deck.
The extra deck is for XYZ, Synchro and Fusion monsters. You guys probably already know about the Fusion monsters, since they were the first and only extra deck monsters you could have. Now there's XYZ's and Synchro cards.
For a Synchro summon, you have to use monster cards who's total level is exactly equal to the level of a Synchro monster that you want to summon. Also, you will need one of the monsters to be a "tuner" monster, which is displayed on the card where it says "Fiend/Tuner" etc. You just have to read what the conditions of the specific Synchro monster are. Some simply require 1 tuner monster and 1 or more non-tuner monsters who's combined level equals the Synchro monsters level.
For example, look at Brionac:
It reads: 1 Tuner + 1 or more non-tuner monsters. So, a level 2 Tuner with a level 4 non-tuner to equal level 6 will be what is required to Synchro summon Brionac. And to do so, you have to have the Tuner and the Non-Tuner on the field, and just send them to the graveyard.
Some Synchro's, however, require additional things not just a tuner and a non-tuner. Some require specific monsters or cards. Some require at least 2 non-tuner monsters with a tuner. It just depends on the card, and the conditions to summon it will be on the text on the card.
XYZ's are a little easier to understand. All you need is 2 or 3 monsters with the same level. Then you overlay them, and Special Summon the XYZ monter on top of the other cards. Some XYZ's require specific types of monsters, however. Such as "2 level 4 warrior type monsters", so pay attention to that.
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