If you get more than one attack per round because your base attack bonus is high enough (see Base Attack Bonus in Classes), because you fight with two weapons or a double weapon, or for some special reason, you must use a full-round action to get your additional attacks.
If a character has the ability to make an attack as a swift, immediate, or free action, are they required to use the full-attack action to do so? Can they move, standard attack, then attack with their swift/immediate/free action?
No, they do not have to take a full-round action because the ability specifies they take the attack as a swift/immediate/free action. This specific text overrides the general rule.
This rule would apply to abilities that add an extra attack without specifying an action. For instance, haste grants you an additional attack but doesn't specify what kind of action it is. Therefor this rule kicks in and you must take a full attack action to use that extra attack.