Smite Evil is limited per day, but not per round. You may use it as many times per round as you like, provided you have enough daily uses remaining. You can use it on any attack you make, regardless of when or why you were making that attack (e.g. any attack in a Full-Attack, a single attack made as a Standard action, an attack of opportunity, attacks from Cleave, so on and so forth). Smite Evil has no impact whatsoever on when or how many attacks you can make; it is an option chosen after deciding to make an attack (a decision which requires that you could make an attack).
There is no direct statement, to my knowledge, that says this, however. It is simply a function of the lack of any statement in the rules about limits per round.
Once per day, a paladin may attempt to smite evil with one normal melee attack. She adds her Charisma bonus (if any) to her attack roll and deals 1 extra point of damage per paladin level. If the paladin accidentally smites a creature that is not evil, the smite has no effect, but the ability is still used up for that day.
At 5th level, and at every five levels thereafter, the paladin may smite evil one additional time per day, as indicated on Table: The Paladin, to a maximum of five times per day at 20th level.
That is the entirety of the rules for Smite Evil. There are no references to per-round limits, or to Swift actions. The “with one normal melee attack” is Smite Evil’s “action,” which is actually what the game refers to as “not an action.”
Not an Action
Some activities are so minor that they are not even considered free actions. They literally don’t take any time at all to do and are considered an inherent part of doing something else.
Smite Evil is an inherent part of attacking, and is thus not an action.
Therefore, Smite Evil is an action that may be taken many times per round, provided you have the daily uses, and does not require any special action.
It definitely does not require a Swift action. The concept of Swift actions didn’t even exist at the time the paladin was printed, having been introduced later on in 3.5.
Finally, since there is apparently some concern about this in your playgroup: Smite Evil is one of the weakest special features in the game (of the core classes’ iconic features, only Favored Enemy is worse). It has extremely tight limits per day, the attack bonus is generally not that large since a paladin needs Strength and Constitution more than he needs Charisma, and the damage bonus is not that impressive. A rogue does dramatically more damage with Sneak Attack if he just gets flanking, and can easily score as many Sneak Attacks in a round as a paladin gets Smite Evil in a day. A barbarian or even a fighter can trivially do more damage than either just by investing in Power Attack and two-handed weapons, and that has no requirements or restrictions. So there is really very little reason to be worried about a paladin saving all his Smite Evils for nuking down one opponent.
Is this different if the smite comes from a weapon property as opposed to a class ability?
That depends on the weapon property; I don’t know of any that do that, so you’d have to give me a book to look in, or just quote the rules text. If it says something like “Smite Evil as the paladin class feature” and doesn’t add any extra qualifications or caveats, then it works exactly the same and everything above remains true, but the item easily could add in extra restrictions relative to the paladin’s version. That would be stated explicitly in the item’s description.