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If the chosen creature is a token, Stunt Double copies the original characteristics of that token as stated by the effect that put the token onto the battlefield. Stunt Double isn’t a token.
If the copied creature has in its mana cost, that X is considered to be 0.
Any enters-the-battlefield abilities of the copied creature will trigger when Stunt Double enters the battlefield. Any “as [this creature] enters the battlefield” or “[this creature] enters the battlefield with” abilities of the chosen creature will also work.
Stunt Double copies exactly what was printed on the original creature (unless that creature is copying something else or is a token; see below). It doesn’t copy whether that creature is tapped or untapped, whether it has any counters on it or any Auras and Equipment attached to it, or any non-copy effects that have changed its power, toughness, types, color, or so on.
If Stunt Double somehow enters the battlefield at the same time as another creature, it can’t become a copy of that creature. You may choose only a creature that’s already on the battlefield.
You can choose not to copy anything. In that case, Stunt Double enters the battlefield as a 0/0 creature, and is put into the graveyard immediately (unless something else is raising its toughness).
If the chosen creature is copying something else (for example, if the chosen creature is another Stunt Double), then Stunt Double enters the battlefield as whatever the chosen creature copied.
