How does a Gemini monster work in Yu-Gi-Oh?

Answer 1

A Gemini monster is a special kind of Effect monster that is treated as a Normal Monster with no effects when initially summoned to the field. At any time after this that you can perform another Normal Summon, you can choose to 'summon it again' (in the OCG, this is called a 'second summon') and it then becomes an
effect monster with the listed effects.

So, Player A, in his main phase, Normal Summons Blazewing Butterfly. It is a Normal Monster, with no effects. In his next turn, he can use his turn's Normal Summon to 'Gemini Summon' the Blazewing Butterfly, which will turn it into an effect monster with the effects on the card. Anything that grants additional Normal Summons can speed this process up.

The card doesn't leave the field at any point, you are performing the summon on a card that is already on the field, and it does not require tribute in the case of a high level Gemini monster. But since it's considered a Normal Summon, it can be negated, and summon-responsive cards like Torrential Tribute can be used against it. Because it never leaves the field, any cards equipped to it, will remain equipped, any bonuses, conditions and counters will be retained also.