From Yu-Gi-Oh!

A Synchro Summon is the act of Special Summoning a Synchro Monster from the Extra Deck using a Tuner monster with additional non-Tuner monsters.

In the anime, while Synchro Summoning, the Tuner monster is said to be tuned with the other monsters.

In the anime, many characters have chants for some of their Synchro Monsters, which they recite while Synchro Summoning them.

How to Summon a Synchro Monster

1. During your Main Phase 1 or 2, when the total Level of a Tuner Monster and 1 or more Non-Tuner Monster(s) you control is equal to the Level of a Synchro Monster you want to summon, you can declare that you are performing a Synchro Summon. These Tuner Monster and the non-Tuner Monster(s) are called the Synchro Material Monsters. Redundant monster(s) are irrelevant.

2. In order to Synchro Summon a Synchro Monster from your Extra Deck, you add the level of a Tuner Monster on your field to the level of another monster(s) on your side of the field. That value must equal the level of the Synchro Monster you wish to summon.

A Synchro Summon is another type of Special Summon.

3. After sending the monsters from your side of the field to the Graveyard, you then place the Synchro Monster from your Extra Deck in face-up Attack Position or Defense Position on your side of the field.

A Level 2 Non- Tuner Monster

A Level 3 Tuner Monster

A Level 5 Synchro Monster

Additional Rules

• A Synchro Summon is one kind of Special Summon, like Fusion Summon and Ritual Summon.
• The mechanics on how a Synchro Summon are done, is quite similar to a Ritual Summon, albeit without the Ritual Spell Card.
• You can Synchro Summon while Macro Cosmos is in play. In this case, the Synchro Material Monsters are removed from play^[1]. Also, Monster Tokens and Trap Monsters (and presumably, Spell Monsters) can be used as Synchro Material Monsters.^[2]
• Synchro Monsters are like Fusion Monsters. Once the Synchro Monster is properly Synchro Summoned, it can be Special Summoned with "Monster Reborn", "Premature Burial", etc. as long as the Synchro Monster is not returned to the Extra Deck. If the Synchro Monster is sent to the Graveyard without being Summoned, such as by the effect of "Gale Dogra", then you cannot Special Summon it.^[3]
• Performing a Synchro Summon does not start a chain.
• You cannot use face-down monsters to perform a Synchro Summon.
• You can negate a Synchro Summon with "Black Horn of Heaven", Horn of Heaven", "Solemn Judgment", "Thunder King Rai-Oh", "Royal Oppression", etc.^[4]
  • In this case, you still send the monsters to the Graveyard. The Synchro Monster is not considered properly Summoned, so you cannot revive it with "Monster Reborn" etc.
• If the Synchro Monster requires Synchro Material Monsters of a specific Attribute (like "Hydro Genex"), you can use a monster with multiple Attributes (like "Elemental Mistress Doriado") as a Synchro Material Monster.^[5]
• The monsters used in Synchro Summoning are not treated as Tributed.
• You may also Synchro Summon more than once per turn, if you have the appropriate monsters out on the field to perform it, since it's a Special Summon.

References

1. ↑ Konami FAQ: [Basic Rules] Can you Synchro Summon a monster when its Synchro Material Monsters would be removed from play?
2. ↑ Konami FAQ: [Basic Rules] Can "Avatar of Apophis", a Monster Token, etc. be used as a Synchro Material Monster?
3. ↑ Konami FAQ: [Basic Rules] Can you Special Summon a Synchro Monster with "Premature Burial" or "D.D.R. - Different Dimension Reincarnation"?
4. ↑ Konami FAQ: [Basic Rules] Can you negate a Synchro Summon with a card effect?
5. ↑ Konami FAQ: [Basic Rules] When Synchro Summoning "Hydro Genex", can you use a monster with multiple Attributes as a Synchro Material Monster?

From Yu-Gi-Oh Card Maker Wiki

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A Synchro Summon is the act of Special Summoning a Synchro Monster from the Extra Deck using a Tuner monster with additional non-Tuner monsters.

In the anime, many characters have chants for some of their Synchro Monsters, which they recite while Synchro Summoning them.

How to Summon a Synchro Monster

1. During your Main Phase 1 or 2, when the total Level of a Tuner Monster and 1 or more Non-Tuner Monster(s) you control is equal to the Level of a Synchro Monster you want to summon, you can declare that you are performing a Synchro Summon. These Tuner Monster and the non-Tuner Monster(s) are called the Synchro Material Monsters. Redundant monster(s) are irrelevant.

2. In order to Synchro Summon a Synchro Monster from your Extra Deck, you add the level of a Tuner Monster on your field to the level of another monster(s) on your side of the field. That value must equal the level of the Synchro Monster you wish to summon.

A Synchro Summon is another type of Special Summon..

3. After sending the monsters from your side of the field to the Graveyard, you then place the Synchro Monster from your Extra Deck in face-up Attack Position or Defense Position on your side of the field.

Additional Rules

• A Synchro Summon is one kind of Special Summon, like Fusion Summon and Ritual Summon
• The mechanics on how a Synchro Summon are done, is quite similar to a Ritual Summon, albeit without the Ritual Spell Card.
• You can Synchro Summon while Macro Cosmos is in play. In this case, the Synchro Material Monsters are removed from play. Also, Monster Tokens and Trap Monsters (and presumably, Spell Monsters) can be used as Synchro Material Monsters.
• Synchro Monsters are like Fusion Monsters. Once the Synchro Monster is properly Synchro Summoned, it can be Special Summoned with "Monster Reborn", "Premature Burial", etc. as long as the Synchro Monster is not returned to the Extra Deck. If the Synchro Monster is sent to the Graveyard without being Summoned, such as by the effect of "Gale Dogra", then you cannot Special Summon it.
• Performing a Synchro Summon does not start a chain.
• You cannot use face-down monsters to perform a Synchro Summon.
• You can negate a Synchro Summon with "Black Horn of Heaven", Horn of Heaven", "Solemn Judgment", "Thunder King Rai-Oh", "Royal Oppression", etc.
  • In this case, you still send the monsters to the Graveyard. The Synchro Monster is not considered properly Summoned, so you cannot revive it with "Monster Reborn" etc.
• If the Synchro Monster requires Synchro Material Monsters of a specific Attribute (like "Hydro Genex"), you can use a monster with multiple Attributes (like "Elemental Mistress Doriado") as a Synchro Material Monster.
• The monsters used in Synchro Summoning are not treated as Tributed.
• You may also Synchro Summon twice per turn if you have the appropriate monsters out on the field to preform it, due to the fact that the rules do not limit you from Special Summoning twice.